International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2010). Vol. E, ch. 1.2, pp. 5-29   | 1 | 2 |
doi: 10.1107/97809553602060000783

Chapter 1.2. Guide to the use of the subperiodic group tables

V. Kopskýa and D. B. Litvinb*

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:  u3c@psu.edu

This chapter forms a guide to the entries of the subperiodic group tables given for the seven crystallographic frieze-group types (two-dimensional groups with one-dimensional translations) in Part 2[link] ; the 75 crystallographic rod-group types (three-dimensional groups with one-dimensional translations) in Part 3[link] ; and the 80 crystallographic layer-group types (three-dimensional groups with two-dimensional translations) in Part 4[link] .

This present volume is, in part, an extension of International Tables for Crystallography, Volume A , Space-Group Symmetry (IT A, 2005[link]). Symmetry tables are given in IT A for the 230 three-dimensional crystallographic space-group types (space groups) and the 17 two-dimensional crystallographic space-group types (plane groups). We give in the following three parts of this volume analogous symmetry tables for the two-dimensional and three-dimensional subperiodic group types: the seven crystallographic frieze-group types (two-dimensional groups with one-dimensional translations) in Part 2[link] ; the 75 crystallographic rod-group types (three-dimensional groups with one-dimensional translations) in Part 3[link] ; and the 80 crystallographic layer-group types (three-dimensional groups with two-dimensional translations) in Part 4[link] . This chapter forms a guide to the entries of the subperiodic group tables given in Parts 2–4.

1.2.1. Classification of subperiodic groups

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Subperiodic groups can be classified in ways analogous to the space groups. For the mathematical definitions of these classifications and their use for space groups, see Chapter 8.2[link] of IT A (2005[link]). Here we shall limit ourselves to those classifications which are explicitly used in the symmetry tables of the subperiodic groups.

1.2.1.1. Subperiodic group types

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The subperiodic groups are classified into affine subperiodic group types, i.e. affine equivalence classes of subperiodic groups. There are 80 affine layer-group types and seven affine frieze-group types. There are 67 crystallographic and an infinity of noncrystallographic (Opechowski, 1986[link]) affine rod-group types. We shall consider here only rod groups of the 67 crystallographic rod-group types. We shall refer here to these crystallographic affine rod-group types simply as affine rod-group types and to the crystallographic rod groups belonging to these types simply as rod groups.

The subperiodic groups are also classified into proper affine subperiodic group types, i.e. proper affine classes of subperiodic groups. For layer and frieze groups, the two classifications are identical. For rod groups, each of eight affine rod-group types splits into a pair of enantiomorphic crystallographic rod-group types. Consequently, there are 75 proper affine rod-group types. The eight pairs of enantiomorphic rod-group types are [{\scr p}4_1] (R24), [{\scr p}4_3] (R26); [{\scr p}4_122] (R31), [{\scr p}4_322] (R33); [{\scr p}3_1] (R43), [{\scr p}3_2] (R44); [{\scr p}3_112] (R47), [{\scr p}3_212] (R48); [{\scr p}6_1] (R54), [{\scr p}6_5] (R58); [{\scr p}6_2] (R55), [{\scr p}6_4] (R57); [{\scr p}6_122] (R63), [{\scr p}6_522] (R67); and [{\scr p}6_222] (R64), [{\scr p}6_422] (R66). (Each subperiodic group is given in the text by its Hermann–Mauguin symbol followed in parenthesis by a letter L, R or F to denote it, respectively, as a layer, rod or frieze group, and its sequential numbering from Parts 2[link] , 3[link] or 4[link] .) We shall refer to the proper affine subperiodic group types simply as subperiodic group types.

1.2.1.2. Other classifications

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There are 27 geometric crystal classes of layer groups and rod groups, and four geometric crystal classes of frieze groups. These are listed, for layer groups, in the fourth column of Table 1.2.1.1[link], and for the rod and frieze groups in the second columns of Tables 1.2.1.2[link] and 1.2.1.3[link], respectively.

Table 1.2.1.1| top | pdf |
Classification of layer groups

Bold or bold underlined symbols indicate Laue groups. Bold underlined point groups are also lattice point symmetries (holohedries).

Two-dimensional Bravais systemSymbolThree-dimensional crystal systemCrystallographic point groupsNo. of layer-group typesRestrictions on conventional coordinate systemCell parameters to be determinedBravais lattice
Oblique m Triclinic 1, [{\bar{\bf 1}}] 2 None a, b, γ mp
    Monoclinic 2, m, 2/m 5 α = β = 90°    
Rectangular o     11 β = γ = 90° a, b op
    Orthorhombic 222, 2mm, mmm 30 α = β = γ = 90°   oc
Square t Tetragonal 4, [\bar{4}], 4/m 16 a = b a tp
      422, 4mm, [\bar{4}]2m, 4/mmm   α = β = γ = 90°    
Hexagonal h Trigonal 3, [{{\bar{\bf 3}}}] 8 a = b a hp
      32, 3m, [{\bar{\bf 3}}{\bi m}]        
    Hexagonal 6, [\bar{6}], 6/m 8 γ = 120°    
      622, 6mm, [\bar{6}]m2, 6/mmm   α = β = 90°    
This angle is conventionally taken to be non-acute, i.e. [\geq]90°.

Table 1.2.1.2| top | pdf |
Classification of rod groups

Bold symbols indicate Laue groups.

Three-dimensional crystal system Crystallographic point groups No. of rod-group types Restrictions on conventional coordinate system
Triclinic 1, [{\bar{\bf 1}}] 2 None
Monoclinic (inclined) 2, m, 2/m 5 β = γ = 90°
Monoclinic (orthogonal)   5 α = β = 90°
Orthorhombic 222, 2mm, mmm 10 α = β = γ = 90°
Tetragonal 4, [\bar{4}], 4/m 19  
  422, 4mm, [\bar{4}]2m, 4/mmm    
Trigonal 3, [{\bar{\bf 3}}] 11 α = β = 90, γ = 120°
  32, 3m, [{\bar{\bf 3}}]m    
Hexagonal 6, [\bar{6}], 6/m 23  
  622, 6mm, [\bar{6}]m2, 6/mmm    

Table 1.2.1.3| top | pdf |
Classification of frieze groups

Bold symbols indicate Laue groups.

Two-dimensional crystal systemCrystallographic point groupsNo. of frieze-group typesRestrictions on conventional coordinate system
Oblique 1, 2 2 None
Rectangular m, 2mm 5 γ = 90°

We further classify subperiodic groups according to the following classifications of the subperiodic group's point group and lattice group. These classifications are introduced to emphasize the relationships between subperiodic groups and space groups:

  • (1) The point group of a layer or rod group is three-dimensional and corresponds to a point group of a three-dimensional space group. The point groups of three-dimensional space groups are classified into the triclinic, monoclinic, ortho­rhombic, tetragonal, trigonal, hexagonal and cubic crystal systems. We shall use this classification also for subperiodic groups. Consequently, the three-dimensional subperiodic groups are classified, see the third column of Table 1.2.1.1[link] and the first column of Table 1.2.1.2[link], into the triclinic, monoclinic, ortho­rhombic, tetragonal, trigonal and hexagonal crystal systems. The cubic crystal system does not arise for three-dimensional subperiodic groups. Two-dimensional subperiodic groups, i.e. frieze groups, are analogously classified, see the first column of Table 1.2.1.3[link], into the oblique and rectangular crystal systems.

  • (2) The two-dimensional lattice of a layer group is also a two-dimensional lattice of a plane group. The lattices of plane groups are classified, according to Bravais (flock) systems, see IT A (2005[link]), into the oblique, rectangular, square and hexagonal Bravais systems. We shall also use this classification for layer groups, see the first column in Table 1.2.1.1[link]. For rod and frieze groups no lattice classification is used, as all one-dimensional lattices form a single Bravais system.

A subdivision of the monoclinic rod-group category is made into monoclinic/inclined and monoclinic/orthogonal. Two different coordinate systems, see Table 1.2.1.2[link], are used for the rod groups of these two subdivisions of the monoclinic crystal system. These two coordinate systems differ in the orientation of the plane containing the non-lattice basis vectors relative to the lattice vectors. For the monoclinic/inclined subdivision, the plane containing the non-lattice basis vectors is, see Fig. 1.2.1.1[link], inclined with respect to the lattice basis vector. For the monoclinic/orthogonal subdivision, the plane is, see Fig. 1.2.1.2[link], orthogonal.

[Figure 1.2.1.1]

Figure 1.2.1.1 | top | pdf |

Monoclinic/inclined basis vectors. For the monoclinic/inclined subdivision, β = γ = 90° and the plane containing the a and b non-lattice basis vectors is inclined with respect to the lattice basis vector c.

[Figure 1.2.1.2]

Figure 1.2.1.2 | top | pdf |

Monoclinic/orthogonal basis vectors. For the monoclinic/orthogonal subdivision, α = β = 90° and the plane containing the a and b non-lattice basis vectors is orthogonal to the lattice basis vector c.

1.2.1.2.1. Conventional coordinate systems

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The subperiodic groups are described by means of a crystallographic coordinate system consisting of a crystallographic origin, denoted by O, and a crystallographic basis. The basis vectors for the three-dimensional layer groups and rod groups are labelled a, b and c. The basis vectors for the two-dimensional frieze groups are labelled a and b. Unlike space groups, not all basis vectors of the crystallographic basis are lattice vectors. Like space groups, the crystallographic coordinate system is used to define the symmetry operations (see Section 1.2.9[link]) and the Wyckoff positions (see Section 1.2.11[link]). The symmetry operations are defined with respect to the directions of both lattice and non-lattice basis vectors. A Wyckoff position, denoted by a coordinate triplet (x, y, z) for the three-dimensional layer and rod groups, is defined in the crystallographic coordinate system by O + r, where r = xa + yb + zc. For the two-dimensional frieze groups, a Wyckoff position is denoted by a coordinate doublet (x, y) and is defined in the crystallographic coordinate system by O + r, where r = xa + yb.

The term setting will refer to the assignment of the labels a, b and c (and the corresponding directions [100], [010] and [001], respectively) to the basis vectors of the crystallographic basis (see Section 1.2.6[link]). In the standard setting, those basis vectors which are also lattice vectors are labelled as follows: for layer groups with their two-dimensional lattice by a and b, for rod groups with their one-dimensional lattice by c, and for frieze groups with their one-dimensional lattice by a.

The selection of a crystallographic coordinate system is not unique. Following IT A (2005[link]), we choose conventional crystallographic coordinate systems which have a right-handed set of basis vectors and such that symmetry of the subperiodic groups is best displayed. The conventional crystallographic coordinate systems used in the standard settings are given in the sixth column of Table 1.2.1.1[link] for the layer groups, and the fourth columns of Tables 1.2.1.2[link] and 1.2.1.3[link] for the rod groups and frieze groups, respectively. The crystallographic origin is conventionally chosen at a centre of symmetry or at a point of high site symmetry (see Section 1.2.7[link]).

The conventional unit cell of a subperiodic group is defined by the crystallographic origin and by those basis vectors which are also lattice vectors. For layer groups in the standard setting, the cell parameters, the magnitude of the lattice basis vectors a and b, and the angle between them, which specify the conventional cell, are given in the seventh column of Table 1.2.1.1[link]. The conventional unit cell obtained in this manner turns out to be either primitive or centred and is denoted by p or c, respectively, in the eighth column of Table 1.2.1.1[link]. For rod and frieze groups with their one-dimensional lattices, the single cell parameter to be specified is the magnitude of the lattice basis vector.

1.2.2. Contents and arrangement of the tables

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The presentation of the subperiodic group tables in Parts 2[link] , 3[link] and 4[link] follows the form and content of IT A (2005[link]). The entries for a subperiodic group are printed on two facing pages or continuously on a single page, where space permits, in the following order (deviations from this standard format are indicated on the relevant pages):

  • Left-hand page:

  • (1) Headline;

  • (2) Diagrams for the symmetry elements and the general position;

  • (3) Origin;

  • (4) Asymmetric unit;

  • (5) Symmetry operations.

  • Right-hand page:

  • (6) Headline in abbreviated form;

  • (7) Generators selected: this information is the basis for the order of the entries under Symmetry operations and Positions;

  • (8) General and special Positions, with the following columns: Multiplicity; Wyckoff letter; Site symmetry, given by the oriented site-symmetry symbol; Coordinates; Reflection conditions;

  • (9) Symmetry of special projections;

  • (10) Maximal non-isotypic non-enantiomorphic subgroups;

  • (11) Maximal isotypic subgroups and enantiomorphic sub­groups of lowest index;

  • (12) Minimal non-isotypic non-enantiomorphic supergroups.

1.2.2.1. Subperiodic groups with more than one description

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For two monoclinic/oblique layer-group types with a glide plane, more than one description is available: p11a (L5) and p112/a (L7). The synoptic descriptions consist of abbreviated treatments for three `cell choices', called `cell choices 1, 2 and 3' [see Section 1.2.6[link], (i) Layer groups[link]]. A complete description is given for cell choice 1 and it is repeated among the synoptic descriptions of cell choices 2 and 3. For three layer groups, p4/n (L52), p4/nbm (L62) and p4/nmm (L64), two descriptions are given (see Section 1.2.7[link]). These two descriptions correspond to the choice of origin, at an inversion centre and on a fourfold axis. For 15 rod-group types, two descriptions are given, corresponding to two settings [see Section 1.2.6[link], (ii) Rod groups[link]].

1.2.3. Headline

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The description of a subperiodic group starts with a headline on a left-hand page, consisting of two or three lines which contain the following information when read from left to right.

  • First line:

    • (1) The short international (Hermann–Mauguin) symbol of the subperiodic group type. Each symbol has two meanings. The first is that of the Hermann–Mauguin symbol of the subperiodic group type. The second meaning is that of a specific subperiodic group which belongs to this subperiodic group type. Given a coordinate system, this group is defined by the list of symmetry operations (see Section 1.2.9[link]) given on the page headed by that Hermann–Mauguin symbol, or by the given list of general positions (see Section 1.2.11[link]). Alternatively, this group is defined by the given diagrams (see Section 1.2.6[link]). The Hermann–Mauguin symbols for the subperiodic group types are distinct except for the rod- and frieze-group types [{\scr p}1] (R1, F1), [{\scr p}211] (R3, F2) and [{\scr p}11m] (R10, F4).

    • (2) The short international (Hermann–Mauguin) point group symbol for the geometric class to which the subperiodic group belongs.

    • (3) The name used in classifying the subperiodic group types. For layer groups this is the combination crystal system/Bravais system classification given in the first two columns of Table 1.2.1.1[link], and for rod and frieze groups this is the crystal system classification in the first columns of Tables 1.2.1.2[link] and 1.2.1.3[link], respectively.

  • Second line:

    • (4) The sequential number of the subperiodic group type.

    • (5) The full international (Hermann–Mauguin) symbol for the subperiodic group type.

    • (6) The Patterson symmetry.

  • Third line:

    This line is used to indicate the cell choice in the case of layer groups p11a (L5) and p112/a (L7), the origin choice for the three layer groups p4/n (L52), p4/nbm (L62) and p4/nmm (L64), and the setting for the 15 rod groups with two distinct Hermann–Mauguin setting symbols (see Table 1.2.6.2[link]).

1.2.4. International (Hermann–Mauguin) symbols for subperiodic groups

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Both the short and the full Hermann–Mauguin symbols consist of two parts: (i) a letter indicating the centring type of the conventional cell, and (ii) a set of characters indicating symmetry elements of the subperiodic group.

  • (i) The letters for the two centring types for layer groups are the lower-case italic letter p for a primitive cell and the lower-case italic letter c for a centred cell. For rod and frieze groups there is only one centring type, the one-dimensional primitive cell, which is denoted by the lower-case script letter [{\scr p}].

  • (ii) The one or three entries after the centring letter refer to the one or three kinds of symmetry directions of the conventional crystallographic basis. Symmetry directions occur either as singular directions or as sets of symmetrically equivalent symmetry directions. Only one representative of each set is given. The sets of symmetry directions and their sequence in the Hermann–Mauguin symbol are summarized in Table 1.2.4.1[link].

    Table 1.2.4.1| top | pdf |
    Sets of symmetry directions and their positions in the Hermann–Mauguin symbol

    (a) Layer groups and rod groups.

     Symmetry direction (position in Hermann–Mauguin symbol)
     PrimarySecondaryTertiary
    Triclinic None
    Monoclinic [100] [010] [001]
    Orthorhombic      
    Tetragonal [001] [[100]] [[1\bar{1}0]]
        [[010]] [[110]]
    Trigonal [001] [[100]] [[1\bar{1}0]]
    Hexagonal   [[010]] [[120]]
        [[\bar{1}\bar{1}0]] [[\bar{2}\bar{1}0]]

    (b) Frieze groups.

     Symmetry direction (position in Hermann–Mauguin symbol)
     PrimarySecondaryTertiary
    Oblique Rotation point in plane    
    Rectangular   [10] [01]

Each position in the Hermann–Mauguin symbol contains one or two characters designating symmetry elements, axes and planes that occur for the corresponding crystallographic symmetry direction. Symmetry planes are represented by their normals; if a symmetry axis and a normal to a symmetry plane are parallel, the two characters are separated by a slash, e.g. the 4/m in [{\scr p}4/mcc] (R40). Crystallographic symmetry directions that carry no symmetry elements are denoted by the symbol `1', e.g. p3m1 (L69) and p112 (L2). If no misinterpretation is possible, entries `1' at the end of the symbol are omitted, as in p4 (L49) instead of p411. Subperiodic groups that have in addition to translations no symmetry directions or only centres of symmetry have only one entry after the centring letter. These are the layer-group types p1 (L1) and [p\bar{1}] (L2), the rod-group types [{\scr p}1] (R1) and [{\scr p}\bar{1}] (R2), and the frieze group [{\scr p}1] (F1).

1.2.5. Patterson symmetry

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The entry Patterson symmetry in the headline gives the subperiodic group of the Patterson function, where Friedel's law is assumed, i.e. with neglect of anomalous dispersion. [For a discussion of the effect of dispersion, see Fischer & Knof (1987[link]) and Wilson (2004[link]).] The symbol for the Patterson subperiodic group can be deduced from the symbol of the subperiodic group in two steps:

  • (i) Glide planes and screw axes are replaced by the corresponding mirror planes and rotation axes.

  • (ii) If the resulting symmorphic subperiodic group is not centrosymmetric, inversion is added.

There are 13 different Patterson symmetries for the layer groups, ten for the rod groups and two for the frieze groups. These are listed in Table 1.2.5.1[link]. The `point-group part' of the symbol of the Patterson symmetry represents the Laue class to which the subperiodic group belongs (cf. Tables 1.2.1.1[link], 1.2.1.2[link] and 1.2.1.3[link]).

Table 1.2.5.1| top | pdf |
Patterson symmetries for subperiodic groups

(a) Layer groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
[\bar{1}] p p[\bar{1}] (L2)
112/m p p112/m (L6)
2/m11 p, c p2/m11 (L14), c2/m11 (L18)
mmm p, c pmmm (L37), cmmm (L47)
4/m p p4/m (L51)
4/mmm p p4/mmm (L61)
[\bar{3}] p p[\bar{3}] (L66)
[\bar{3}]1m p p[\bar{3}]1m (L71)
[\bar{3}]m1 p p[\bar{3}]m1 (L72)
6/m p p6/m (L75)
6/mmm p p6/mmm (L80)

(b) Rod groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
[\bar{1}] [{\scr p}] [\scr p][\bar{1}] (R2)
2/m11 [{\scr p}] [\scr p]2/m11 (R6)
112/m [{\scr p}] [\scr p]112/m (R11)
mmm [{\scr p}] [\scr p]mmm (R20)
4/m [{\scr p}] [\scr p]4/m (R28)
4/mmm [{\scr p}] [\scr p]4/mmm (R39)
[\bar{3}] [{\scr p}] [\scr p][\bar{3}] (R48)
[\bar{3}]m [{\scr p}] [\scr p][\bar{3}]1m (R51)
6/m [{\scr p}] [\scr p]6/m (R60)
6/mmm [{\scr p}] [\scr p]6/mmm (R73)

(c) Frieze groups.

Laue classLattice typePatterson symmetry (with subperiodic group number)
2 [{\scr p}] [\scr p]211 (F2)
2mm [{\scr p}] [\scr p]2mm (F6)

1.2.6. Subperiodic group diagrams

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There are two types of diagrams, referred to as symmetry diagrams and general-position diagrams. Symmetry diagrams show (i) the relative locations and orientations of the symmetry elements and (ii) the locations and orientations of the symmetry elements relative to a given coordinate system. General-position diagrams show the arrangement of a set of symmetrically equivalent points of general positions relative to the symmetry elements in that given coordinate system.

For the three-dimensional subperiodic groups, i.e. layer and rod groups, all diagrams are orthogonal projections. The projection direction is along a basis vector of the conventional crystallographic coordinate system (see Tables 1.2.1.1[link] and 1.2.1.2[link]). If the other basis vectors are not parallel to the plane of the figure, they are indicated by subscript `p', e.g. ap, bp and cp. For frieze groups (two-dimensional subperiodic groups), the diagrams are in the plane defined by the frieze group's conventional crystallographic coordinate system (see Table 1.2.1.3[link]).

The graphical symbols for symmetry elements used in the symmetry diagrams are given in Chapter 1.1[link] and follow those used in IT A (2005[link]). For rod groups, the `heights' h along the projection direction above the plane of the diagram are indicated for symmetry planes and symmetry axes parallel to the plane of the diagram, for rotoinversions and for centres of symmetry. The heights are given as fractions of the translation along the projection direction and, if different from zero, are printed next to the graphical symbol.

Schematic representations of the diagrams, displaying their conventional coordinate system, i.e. the origin and basis vectors, with the basis vectors labelled in the standard setting, are given below. The general-position diagrams are indicated by the letter [{\sf G}].

  • (i) Layer groups

    For the layer groups, all diagrams are orthogonal projections along the basis vector c. For the triclinic/oblique layer groups, two diagrams are given: the general-position diagram on the right and the symmetry diagram on the left. These diagrams are illustrated in Fig. 1.2.6.1[link].

    [Figure 1.2.6.1]

    Figure 1.2.6.1 | top | pdf |

    Diagrams for triclinic/oblique layer groups.

    For all monoclinic/oblique layer groups, except groups L5 and L7, two diagrams are given, as shown in Fig. 1.2.6.2[link]. For the layer groups L5 and L7, the descriptions of the three cell choices are headed by a pair of diagrams, as illustrated in Fig. 1.2.6.3[link]. Each diagram is a projection of four neighbouring unit cells. The headline of each cell choice contains a small drawing indicating the origin and basis vectors of the cell that apply to that description.

    [Figure 1.2.6.2]

    Figure 1.2.6.2 | top | pdf |

    Diagrams for monoclinic/oblique layer groups.

    [Figure 1.2.6.3]

    Figure 1.2.6.3 | top | pdf |

    Monoclinic/oblique layer groups Nos. 5 and 7, cell choices 1, 2, 3. The numbers 1, 2, 3 within the cells and the subscripts of the basis vectors indicate the cell choice.

    For the monoclinic/rectangular and orthorhombic/rectangular layer groups, two diagrams are given, as illustrated in Figs. 1.2.6.4[link] and 1.2.6.5[link], respectively. For these groups, the Hermann–Mauguin symbol for the layer group is given for two settings, i.e. for two ways of assigning the labels a, b, c to the basis vectors of the conventional coordinate system.

    [Figure 1.2.6.4]

    Figure 1.2.6.4 | top | pdf |

    Diagrams for monoclinic/rectangular layer groups.

    [Figure 1.2.6.5]

    Figure 1.2.6.5 | top | pdf |

    Diagrams for orthorhombic/rectangular layer groups.

    The symbol for each setting is referred to as a setting symbol. The setting symbol for the standard setting is ([abc]). The Hermann–Mauguin symbol of the layer group in the conventional coordinate system, in the standard setting, is the same as the Hermann–Mauguin symbol in the first line of the headline. The setting symbol for all other settings is a shorthand notation for the relabelling of the basis vectors. For example, the setting symbol ([cab]) means that the basis vectors relabelled in this setting as a, b and c were in the standard setting labelled c, a and b, respectively [cf. Section 2.2.6[link] of IT A (2005)].

    For these groups, the two settings considered are the standard ([abc]) setting and a second ([b{\bar a}c]) setting. In Fig. 1.2.6.6[link], the ([abc]) setting symbol is written horizontally across the top of the diagram and the second ([b{\bar a}c]) setting symbol is written vertically on the left-hand side of the diagram. When viewing the diagram with the ([abc]) setting symbol written horizontally across the top of the diagram, the origin of the coordinate system is at the upper left-hand corner of the diagram, the basis vector labelled a is downward towards the bottom of the page, the basis vector labelled b is to the right and the basis vector labelled c is upward out of the page (see also Figs. 1.2.6.4[link] and 1.2.6.5[link]). When viewing the diagram with the ([b{\bar a}c]) written horizontally, i.e. by rotating the page clockwise by 90° or by viewing the diagram from the right, the position of the origin and the labelling of the basis vectors are as above, i.e. the origin is at the upper left-hand corner, the basis vector labelled a is downward, the basis vector labelled b is to the right and the basis vector labelled c is upward out of the page. In the symmetry diagrams of these groups, Part 4[link] , the setting symbols are not given. In their place is given the Hermann–Mauguin symbol of the layer group in the conventional coordinate system in the corresponding setting. The Hermann–Mauguin symbol in the standard setting is given horizontally across the top of the diagram, and in the second setting vertically on the left-hand side.

    [Figure 1.2.6.6]

    Figure 1.2.6.6 | top | pdf |

    Monoclinic/rectangular and orthorhombic/rectangular layer groups with two settings. For the second-setting symbol printed vertically, the page must be turned clockwise by 90° or viewed from the right-hand side.

    If the two Hermann–Mauguin symbols are the same (i.e. as the Hermann–Mauguin symbol in the first line of the heading), then no symbols are explicitly given. A listing of monoclinic/rectangular and orthorhombic/rectangular layer groups with distinct Hermann–Mauguin symbols in the two settings is given in Table 1.2.6.1[link].

    Table 1.2.6.1| top | pdf |
    Distinct Hermann–Mauguin symbols for monoclinic/rectangular and orthorhombic/rectangular layer groups in different settings

    Layer group Setting symbol
    (abc) ([b\bar{a}c])
    Hermann–Mauguin symbol
    L8 p211 p121
    L9 p2111 p1211
    L10 c211 c121
    L11 pm11 p1m1
    L12 pb11 p1a1
    L13 cm11 c1m1
    L14 p2/m11 p12/m1
    L15 p21/m11 p121/m1
    L16 p2/b11 p12/a1
    L17 p21/b11 p121/a1
    L18 c2/m11 c12/m1
    L20 p2122 p2212
    L24 pma2 pbm2
    L27 pm2m p2mm
    L28 pm21b p21ma
    L29 pb21m p21am
    L30 pb2b p2aa
    L31 pm2a p2mb
    L32 pm21n p21mn
    L33 pb21a p21ab
    L34 pb2n p2an
    L35 cm2m c2mm
    L36 cm2a c2mb
    L38 pmaa pbmb
    L40 pmam pbmm
    L41 pmma pmmb
    L42 pman pbmn
    L43 pbaa pbab
    L45 pbma pmab

    Example: The layer group [pma2] (L24)

    In the ([abc]) setting, the Hermann–Mauguin symbol is pma2. In the ([b{\bar a}c]) setting, the Hermann–Mauguin symbol is pbm2.

    For the square/tetragonal, hexagonal/trigonal and hexagonal/hexagonal layer groups, two diagrams are given, as illustrated in Figs. 1.2.6.7[link] and 1.2.6.8[link].

    [Figure 1.2.6.7]

    Figure 1.2.6.7 | top | pdf |

    Diagrams for square/tetragonal layer groups.

    [Figure 1.2.6.8]

    Figure 1.2.6.8 | top | pdf |

    Diagrams for trigonal/hexagonal and hexagonal/hexagonal layer groups.

  • (ii) Rod groups

    For triclinic, monoclinic/inclined, monoclinic/orthogonal and orthorhombic rod groups, six diagrams are given: three symmetry diagrams and three general-position diagrams. These diagrams are orthogonal projections along each of the conventional coordinate system basis vectors. For pictorial clarity, each of the projections contains an area bounded by a circle or a parallelogram. These areas may be considered as the projections of a cylindrical volume, whose axis coincides with the c lattice vector, bounded at [z = 0] and [z = 1] by planes parallel to the plane containing the a and b basis vectors. The projection of the c lattice vector is shown explicitly. Only the directions of the projected non-lattice basis vectors a and b are indicated in the diagrams, denoted by lines from the origin to the boundary of the projected cylinder. These diagrams are illustrated for triclinic rod groups in Fig. 1.2.6.9[link], for monoclinic/inclined rod groups in Fig. 1.2.6.10[link], for monoclinic/orthogonal rod groups in Fig. 1.2.6.11[link] and for orthorhombic rod groups in Fig. 1.2.6.12[link].

    [Figure 1.2.6.9]

    Figure 1.2.6.9 | top | pdf |

    Diagrams for triclinic rod groups.

    [Figure 1.2.6.10]

    Figure 1.2.6.10 | top | pdf |

    Diagrams for monoclinic/inclined rod groups.

    [Figure 1.2.6.11]

    Figure 1.2.6.11 | top | pdf |

    Diagrams for monoclinic/orthogonal rod groups.

    [Figure 1.2.6.12]

    Figure 1.2.6.12 | top | pdf |

    Diagrams for orthorhombic rod groups.

    The symmetry diagrams consist of the c projection, outlined with a circle at the upper left-hand side, the a projection at the lower left-hand side and the b projection at the upper right-hand side. The general-position diagrams are the c projection, outlined with a circle at the lower right-hand side, and the remaining two general-position diagrams next to the corresponding symmetry diagrams.

    Six settings for each of these rod groups are considered and the corresponding setting symbols are shown in Fig. 1.2.6.13[link]. This figure schematically shows the three symmetry diagrams each with two setting symbols, one written horizontally across the top of the diagram and the second written vertically along the left-hand side of the diagram. In the symmetry diagrams of these groups, Part 3[link] , the setting symbols are not given. In their place is given the Hermann–Mauguin symbol of the layer group in the conventional coordinate system in the corresponding setting. As there are only translations in one dimension, it is necessary to add to the translational part of the Hermann–Mauguin symbol a subindex to the lattice symbol to denote the direction of the translations. For example, consider the rod group of the type [{\scr p}211] (R3). The Hermann–Mauguin symbol in the conventional coordinate system in the standard ([abc]) setting is given by [{\scr p}_c211] as the translations of the rod group in the standard setting are along the direction labelled c. In the ([bca]) setting, the Hermann–Mauguin symbol is [{\scr p}_b112], where the subindex b denotes that the translations are, in this setting, along the direction labelled b. A list of the six Hermann–Mauguin symbols in the six settings for the triclinic, monoclinic/inclined, monoclinic/orthogonal and ortho­rhombic rod groups is given in Table 1.2.6.2[link].

    Table 1.2.6.2| top | pdf |
    Distinct Hermann–Mauguin symbols for monoclinic and orthorhombic rod groups in different settings

    Rod groupSetting symbol
    (abc)([b\bar{a}c])([\bar{c}ba])(bca)([a\bar{c}b])([\bar{c}\bar{a}b])
    Hermann–Mauguin symbol
    R3 [\scr p]c211 [\scr p]c121 [\scr p]a112 [\scr p]b112 [\scr p]b211 [\scr p]a121
    R4 [\scr p]cm11 [\scr p]c1m1 [\scr p]a11m [\scr p]b11m [\scr p]bm11 [\scr p]a1m1
    R5 [\scr p]cc11 [\scr p]c1c1 [\scr p]a11a [\scr p]b11b [\scr p]bb11 [\scr p]a1a1
    R6 [\scr p]c2/m11 [\scr p]c12/m1 [\scr p]a112/m [\scr p]b112/m [\scr p]b2/m11 [\scr p]a12/m1
    R7 [\scr p]c2/c11 [\scr p]c12/c1 [\scr p]a112/a [\scr p]b112/b [\scr p]b2/b11 [\scr p]a12/a1
    R8 [\scr p]c112 [\scr p]c112 [\scr p]a211 [\scr p]b121 [\scr p]b121 [\scr p]a211
    R9 [\scr p]c1121 [\scr p]c1121 [\scr p]a2111 [\scr p]b1211 [\scr p]b1211 [\scr p]a2111
    R10 [\scr p]c11m [\scr p]c11m [\scr p]am11 [\scr p]b1m1 [\scr p]b1m1 [\scr p]am11
    R11 [\scr p]c112/m [\scr p]c112/m [\scr p]a2/m11 [\scr p]b12/m1 [\scr p]b12/m1 [\scr p]a2/m11
    R12 [\scr p]c1121/m [\scr p]c1121/m [\scr p]a21/m11 [\scr p]b121/m1 [\scr p]b121/m1 [\scr p]a21/m11
    R13 [\scr p]c222 [\scr p]c222 [\scr p]a222 [\scr p]b222 [\scr p]b222 [\scr p]a222
    R14 [\scr p]c2221 [\scr p]c2221 [\scr p]a2122 [\scr p]b2212 [\scr p]b2212 [\scr p]a2122
    R15 [\scr p]cmm2 [\scr p]cmm2 [\scr p]a2mm [\scr p]bm2m [\scr p]bm2m [\scr p]a2mm
    R16 [\scr p]ccc2 [\scr p]ccc2 [\scr p]a2aa [\scr p]bb2b [\scr p]bb2b [\scr p]a2aa
    R17 [\scr p]cmc21 [\scr p]ccm21 [\scr p]a21am [\scr p]bb21m [\scr p]bm21b [\scr p]a21ma
    R18 [\scr p]c2mm [\scr p]cm2m [\scr p]amm2 [\scr p]bmm2 [\scr p]b2mm [\scr p]am2m
    R19 [\scr p]c2cm [\scr p]cc2m [\scr p]ama2 [\scr p]bbm2 [\scr p]b2mb [\scr p]am2a
    R20 [\scr p]cmmm [\scr p]cmmm [\scr p]ammm [\scr p]bmmm [\scr p]bmmm [\scr p]ammm
    R21 [\scr p]cccm [\scr p]cccm [\scr p]amaa [\scr p]bbmb [\scr p]bbmb [\scr p]amaa
    R22 [\scr p]cmcm [\scr p]ccmm [\scr p]amam [\scr p]bbmm [\scr p]bmmb [\scr p]amma
    [Figure 1.2.6.13]

    Figure 1.2.6.13 | top | pdf |

    Setting symbols on symmetry diagrams for the monoclinic/inclined, monoclinic/orthogonal and orthorhombic rod groups.

    Example: The rod group [{\scr p}mc2_1] (R17)

    The Hermann–Mauguin setting symbols for the six settings are:[\matrix{\hbox{Setting symbol}\hfill &\hbox{Hermann$-$Mauguin symbol}\hfill\cr (abc)\hfill & {\scr p}_cmc2_1\hfill\cr (b{\bar a}c)\hfill & {\scr p}_ccm2_1\hfill\cr ({\bar c}ba)\hfill &{\scr p}_a2_1am\hfill\cr (bca)\hfill &{\scr p}_bb2_1m\hfill\cr (a{\bar c}b)\hfill &{\scr p}_bm2_1b\hfill\cr ({\bar c}{\bar a}b)\hfill & {\scr p}_a2_1ma\hfill}]

    For tetragonal, trigonal and hexagonal rod groups, two diagrams are given: the symmetry diagram and the general-position diagram. These diagrams are illustrated in Figs. 1.2.6.14[link] and 1.2.6.15[link]. One can consider additional settings for these rod groups: see the setting symbols in Table 1.2.6.3[link]. If the Hermann–Mauguin symbols for the group in these settings are identical, only one tabulation of the group, in the standard setting, is given. If in these settings two distinct Hermann–Mauguin symbols are obtained, a second tabulation for the rod group is given. This second tabulation is in the conventional coordinate system in the ([a+b\;\;{\overline a}+b\;\;c]) setting for tetragonal groups, and in the ([2a+b\;\; \overline{a}+b\;\;c]) setting for trigonal and hexagonal groups. These second tabulations aid in the correlation of Wyckoff positions of space groups and Wyckoff positions of rod groups. For example, the Wyckoff positions of the two space groups types P3m1 and P31m can be easily correlated with, respectively, the Wyckoff positions of a rod group of the type R49 in the standard setting where the Hermann–Mauguin symbol is [{\scr p}3m1] and in the second setting where the symbol is [{\scr p}31m]. In Table 1.2.6.3[link], we list the tetragonal, trigonal and hexagonal rod groups where in the different settings the two Hermann–Mauguin symbols are distinct.

    Table 1.2.6.3| top | pdf |
    Distinct Hermann–Mauguin symbols for tetragonal, trigonal and hexagonal rod groups in different settings

    Rod groupSetting symbol
    [(abc)][(a\pm b\ \ b\mp a\ \ c)]
    Hermann–Mauguin symbol
    R35 [\scr p]42cm [\scr p]42mc
    R37 [\scr p][\bar{4}]2m [\scr p][\bar{4}]m2
    R38 [\scr p][\bar{4}]2c [\scr p][\bar{4}]c2
    R41 [\scr p]42/mmc [\scr p]42/mcm

    Rod groupSetting symbol
    [(abc)][\matrix{(\pm 2a\pm b\ \ \ \mp\! a\pm b\ \ \ c)\hfill\cr (\pm a\pm 2b\ \ \ \mp\! 2a\mp b \ \ \ c)\hfill\cr (\mp a\pm b\ \ \ \mp\! a\mp 2b\ \ \ c)\hfill\cr}]
    Hermann–Mauguin symbol
    R46 [\scr p]312 [\scr p]321
    R47 [\scr p]3112 [\scr p]3121
    R48 [\scr p]3212 [\scr p]3221
    R49 [\scr p]3m1 [\scr p]31m
    R50 [\scr p]3c1 [\scr p]31c
    R51 [\scr p][\bar{3}]1m [\scr p][\bar{3}]m1
    R52 [\scr p][\bar{3}]1c [\scr p][\bar{3}]c1
    R70 [\scr p]63mc [\scr p]63cm
    R71 [\scr p][\bar{6}]m2 [\scr p][\bar{6}]2m
    R72 [\scr p][\bar{6}]c2 [\scr p][\bar{6}]2c
    R75 [\scr p]63/mmc [\scr p]63/mcm
    [Figure 1.2.6.14]

    Figure 1.2.6.14 | top | pdf |

    Diagrams for tetragonal rod groups.

    [Figure 1.2.6.15]

    Figure 1.2.6.15 | top | pdf |

    Diagrams for trigonal and hexagonal rod groups.

  • (iii) Frieze groups

    Two diagrams are given for each frieze group: a symmetry diagram and a general-position diagram. These diagrams are illustrated for the oblique and rectangular frieze groups in Figs. 1.2.6.16[link] and 1.2.6.17[link], respectively. We consider the two settings (ab) and ([{b\overline{a}}]), see Fig. 1.2.6.18[link]. In the frieze-group tables, Part 2[link] , we replace the setting symbols with the corresponding Hermann–Mauguin symbols where a subindex is added to the lattice symbol to denote the direction of the translations. A listing of the frieze groups with the Hermann–Mauguin symbols of each group in the two settings is given in Table 1.2.6.4[link].

    Table 1.2.6.4| top | pdf |
    Distinct Hermann–Mauguin symbols for frieze groups in different settings

     Setting symbol
     (ab)([b\bar{a}])
    Frieze groupHermann–Mauguin symbol
    F1 [\scr p]a1 [\scr p]b1
    F2 [\scr p]a211 [\scr p]b211
    F3 [\scr p]a1m1 [\scr p]b11m
    F4 [\scr p]a11m [\scr p]b1m1
    F5 [\scr p]a11g [\scr p]b1g1
    F6 [\scr p]a2mm [\scr p]b2mm
    F7 [\scr p]a2mg [\scr p]b2gm
    [Figure 1.2.6.16]

    Figure 1.2.6.16 | top | pdf |

    Diagrams for oblique frieze groups.

    [Figure 1.2.6.17]

    Figure 1.2.6.17 | top | pdf |

    Diagrams for rectangular frieze groups.

    [Figure 1.2.6.18]

    Figure 1.2.6.18 | top | pdf |

    The two settings for frieze groups. For the second setting, printed vertically, the page must be turned 90° clockwise or viewed from the right-hand side.

1.2.7. Origin

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The origin has been chosen according to the following conventions:

  • (i) If the subperiodic group is centrosymmetric, then the inversion centre is chosen as the origin. For the three layer groups p4/n (L52), p4/nbm (L62) and p4/nmm (L64), we give descriptions for two origins, at the inversion centre and at ([-{1 \over 4}, -{1 \over 4}, 0]) from the inversion centre. This latter origin is at a position of high site symmetry and is consistent with having the origin on the fourfold axis, as is the case for all other tetragonal layer groups. The group symbols for the description with the origin at the inversion centre, e.g. [p4/n\, (\,{1 \over 4},{1 \over 4},0)], are followed by the shift [(\,{1 \over 4},{1 \over 4},0)] of the position of the origin used in the description having the origin on the fourfold axis.

  • (ii) For noncentrosymmetric subperiodic groups, the origin is at a point of highest site symmetry. If no symmetry is higher than 1, the origin is placed on a screw axis, a glide plane or at the intersection of several such symmetry elements.

Origin statement: In the line Origin immediately below the diagrams, the site symmetry of the origin is stated if different from the identity. A further symbol indicates all symmetry elements that pass through the origin. For the three layer groups p4/n (L52), p4/nbm (L62) and p4/nmm (L64) where the origin is on the fourfold axis, the statement `at [-{1 \over 4}, -{1 \over 4}, 0] from centre' is given to denote the position of the origin with respect to an inversion centre.

1.2.8. Asymmetric unit

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An asymmetric unit of a subperiodic group is a simply connected smallest part of space from which, by application of all symmetry operations of the subperiodic group, the whole space is filled exactly. For three-dimensional (two-dimensional) space groups, because they contain three-dimensional (two-dimensional) translational symmetry, the asymmetric unit is a finite part of space [see Section 2.2.8[link] of IT A (2005[link])]. For subperiodic groups, because the translational symmetry is of a lower dimension than that of the space, the asymmetric unit is infinite in size. We define the asymmetric unit for subperiodic groups by setting the limits on the coordinates of points contained in the asymmetric unit.

1.2.8.1. Frieze groups

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For all frieze groups, a limit is set on the x coordinate of the asymmetric unit by the inequality[0\leq x \leq \hbox{ upper limit on }x.]For the y coordinate, either there is no limit and nothing further is written, or there is the lower limit of zero, i.e. [0\leq y].

Example: The frieze group [{\scr p}2mm] (F6)

[\displaylines{\quad{\bf Asymmetric}\,\,{\bf unit}\quad 0\leq x\leq {1/2}; \,0\leq y.\hfill}]

1.2.8.2. Rod groups

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For all rod groups, a limit is set on the z coordinate of the asymmetric unit by the inequality[0\leq z\leq\hbox{ upper limit on }z.]For each of the x and y coordinates, either there is no limit and nothing further is written, or there is the lower limit of zero.

For tetragonal, trigonal and hexagonal rod groups, additional limits are required to define the asymmetric unit. These limits are given by additional inequalities, such as [x\leq y] and [y\leq x/2]. Fig. 1.2.8.1[link] schematically shows the boundaries represented by such inequalities.

[Figure 1.2.8.1]

Figure 1.2.8.1 | top | pdf |

Boundaries used to define the asymmetric unit for (a) tetragonal rod groups and (b) trigonal and hexagonal rod groups.

Example: The rod group [{\scr p}6_3mc] (R70)

[\displaylines{\quad{\bf Asymmetric}\,\,{\bf unit}\quad 0\leq x;\,0\leq y;\,0\leq z\leq 1;\,y\leq x/2.\hfill}]

1.2.8.3. Layer groups

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For all layer groups, limits are set on the x coordinate and y coordinate of the asymmetric unit by the inequalities[\eqalign{0\leq x\leq&\hbox{ upper limit on }x\cr 0\leq y\leq&\hbox{ upper limit on }y.}]For the z coordinate, either there is no limit and nothing further is written, or there is the lower limit of zero.

For tetragonal/square, trigonal/hexagonal and hexagonal/hexagonal layer groups, additional limits are required to define the asymmetric unit. These additional limits are given by additional inequalities. Fig. 1.2.8.2[link] schematically shows the boundaries represented by these inequalities. For trigonal/hexagonal and hexagonal/hexagonal layer groups, because of the complicated shape of the asymmetric unit, the coordinates (x, y) of the vertices of the asymmetric unit with the z = 0 plane are given.

[Figure 1.2.8.2]

Figure 1.2.8.2 | top | pdf |

Boundaries used to define the asymmetric unit for (a) tetragonal/square layer groups and (b) trigonal/hexagonal and hexagonal/hexagonal layer groups. In (b), the coordinates (x, y) of the vertices of the asymmetric unit with the z = 0 plane are also given.

Example: The layer group [p3m1] (L69)

[\displaylines{{\bf Asymmetric}\,\,{\bf unit}\quad 0\leq x\leq 2/3;\,0\leq y \leq 2/3;\,x\leq 2y;\hfill\cr\phantom{{\bf Asymmetric}\,\,{\bf unit}\quad}y\leq{\rm min}\,(1-x,2x)\hfill \cr{\bf Vertices}\quad 0,0;\,\,2/3,1/3;\,\,1/3,2/3.\hfill}]

1.2.9. Symmetry operations

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The coordinate triplets of the General position of a subperiodic group may be interpreted as a shorthand description of the symmetry operations in matrix notation as in the case of space groups [see Sections 2.2.3[link] , 8.1.5[link] and 11.1.1[link] of IT A (2005[link])]. The geometric description of the symmetry operations is found in the subperiodic group tables under the heading Symmetry operations. These data form a link between the subperiodic group diagrams (Section 1.2.6[link]) and the general position (Section 1.2.11[link]). Below the geometric description we give the Seitz notation (Burns & Glazer, 1990[link]) of each symmetry operation using the subindex notation of Zak et al. (1969[link]).

1.2.9.1. Numbering scheme

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The numbering [(1)\,\ldots\, (\,p)\,\ldots] of the entries in the blocks Symmetry operations and General position (first block below Positions) is the same. Each listed coordinate triplet of the general position is preceded by a number between parentheses (p). The same number (p) precedes the corresponding symmetry operation. For all subperiodic groups with primitive lattices, the two lists contain the same number of entries.

For the nine layer groups with centred lattices, to the one block of General positions correspond two blocks of Symmetry operations. The numbering scheme is applied to both blocks. The two blocks correspond to the two centring translations below the subheading Coordinates, i.e. [(0, 0, 0)\!+ \;\;(1/2, 1/2, 0)+]. For the Positions, the reader is expected to add these two centring translations to each printed coordinate triplet in order to obtain the complete general position. For the Symmetry operations, the corresponding data are listed explicitly with the two blocks having the subheadings `For (0, 0, 0)+ set' and `For (1/2, 1/2, 0)+ set', respectively.

1.2.9.2. Designation of symmetry operations

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The designation of symmetry operations for the subperiodic groups is the same as for the space groups. An entry in the block Symmetry operations is characterized as follows:

  • (i) A symbol denoting the type of the symmetry operation [cf. Chapter 1.2[link] of IT A (2005[link])], including its glide or screw part, if present. In most cases, the glide or screw part is given explicitly by fractional coordinates between parentheses. The sense of a rotation is indicated by the superscript + or −. Abbreviated notations are used for the glide reflections a(1/2, 0, 0) ≡ a; b(0, 1/2, 0) ≡ b; c(0, 0, 1/2) ≡ c. Glide reflections with complicated and unconventional glide parts are designated by the letter g, followed by the glide part between parentheses.

  • (ii) A coordinate triplet indicating the location and orientation of the symmetry element which corresponds to the symmetry operation. For rotoinversions the location of the inversion point is also given.

Details of this symbolism are given in Section 11.1.2[link] of IT A (2005[link]).

Examples

  • (1) [m\quad x, 0, z]: a reflection through the plane [x, 0, z], i.e. the plane parallel to (010) containing the point (0, 0, 0).

  • (2) [m\quad x+1/2, \bar{x}, z]: a reflection through the plane [x+1/2, \bar{x}, z], i.e. the plane parallel to (110) containing the point (1/2, 0, 0).

  • (3) [g(1/2, 1/2 ,0)\quad x,x,z]: glide reflection with glide component (1/2, 1/2, 0) through the plane [x, x, z], i.e. the plane parallel to ([1\bar{1}0]) containing the point (0, 0, 0).

  • (4) [2(1/2,0,0)\quad x,1/4,0]: screw rotation along the (100) direction containing the point (0, 1/4, 0) with a screw component (1/2, 0, 0).

  • (5) [\bar{4}^{-}\quad 1/2,0,z\quad 1/2,0,0]: fourfold rotoinversion consisting of a clockwise rotation by 90° around the line 1/2, 0, z followed by an inversion through the point (1/2, 0, 0).

1.2.10. Generators

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The line Generators selected states the symmetry operations and their sequence selected to generate all symmetrically equivalent points of the General position from a point with coordinates [x,y,z]. The identity operation given by (1) is always selected as the first generator. The generating translations are listed next, t(1, 0) for frieze groups, t(0, 0, 1) for rod groups, and t(1, 0, 0) and t(0, 1, 0) for layer groups. For centred layer groups, there is the additional centring translation t(1/2, 1/2, 0). The additional generators are given as numbers (p) which refer to the corresponding coordinate triplets of the general position and the corresponding entries under Symmetry operations; for centred layer groups, the first block `For (0, 0, 0)+ set' must be used.

1.2.11. Positions

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The entries under Positions (more explicitly called Wyckoff positions) consist of the General position (upper block) and the Special positions (blocks below). The columns in each block, from left to right, contain the following information for each Wyckoff position.

  • (i) Multiplicity M of the Wyckoff position. This is the number of equivalent points per conventional cell. The multiplicity M of the general position is equal to the order of the point group of the subperiodic group, except in the case of centred layer groups when it is twice the order of the point group. The multiplicity M of a special position is equal to the order of the point group of the subperiodic group divided by the order of the site-symmetry group (see Section 1.2.12[link]).

  • (ii) Wyckoff letter. This letter is a coding scheme for the Wyckoff positions, starting with a at the bottom position and continuing upwards in alphabetical order.

  • (iii) Site symmetry. This is explained in Section 1.2.12[link].

  • (iv) Coordinates. The sequence of the coordinate triplets is based on the Generators. For the centred layer groups, the centring translations (0, 0, 0)+ and (1/2, 1/2, 0)+ are listed above the coordinate triplets. The symbol `+' indicates that in order to obtain a complete Wyckoff position, the components of these centring translations have to be added to the listed coordinate triplets.

  • (v) Reflection conditions. These are described in Section 1.2.13[link].

The two types of positions, general and special, are characterized as follows:

  • (i) General position. A set of symmetrically equivalent points is said to be in a `general position' if each of its points is left invariant only by the identity operation but by no other symmetry operation of the subperiodic group.

  • (ii) Special position(s). A set of symmetrically equivalent points is said to be in a `special position' if each of its points is mapped onto itself by at least one additional operation in addition to the identity operation.

Example: Layer group [c2/m11] (L18)

The general position 8f of this layer group contains eight equivalent points per cell each with site symmetry 1. The coordinate triplets of four points (1) to (4) are given explicitly, the coordinate triplets of the other four points are obtained by adding the components (1/2, 1/2, 0) of the c-centring translation to the coordinate triplets (1) to (4).

This layer group has five special positions with the Wyckoff letters a to e. The product of the multiplicity and the order of the site-symmetry group is the multiplicity of the general position. For position 4d, for example, the four equivalent points have the coordinates [x,0,0], [\bar{x},0,0], [x+1/2,1/2,0] and [\bar{x}+1/2,1/2,0]. Since each point of position 4d is mapped onto itself by a twofold rotation, the multiplicity of the position is reduced from eight to four, whereas the order of the site symmetry is increased from one to two.

1.2.12. Oriented site-symmetry symbols

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The third column of each Wyckoff position gives the site symmetry of that position. The site-symmetry group is isomorphic to a proper or improper subgroup of the point group to which the subperiodic group under consideration belongs. Oriented site-symmetry symbols are used to show how the symmetry elements at a site are related to the conventional crystallographic basis. The site-symmetry symbols display the same sequence of symmetry directions as the subperiodic group symbol (cf. Table 1.2.4.1[link]). Sets of equivalent symmetry directions that do not contribute any element to the site-symmetry group are represented by a dot. Sets of symmetry directions having more than one equivalent direction may require more than one character if the site-symmetry group belongs to a lower crystal system. For example, for the 2c position of tetragonal layer group p4mm (L55), the site-symmetry group is the orthorhombic group `2mm.'. The two characters `mm' represent the secondary set of tetragonal symmetry directions, whereas the dot represents the tertiary tetragonal symmetry direction.

1.2.13. Reflection conditions

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The Reflection conditions are listed in the right-hand column of each Wyckoff position. There are two types of reflection conditions:

  • (i) General conditions. These conditions apply to all Wyckoff positions of the subperiodic group.

  • (ii) Special conditions (`extra' conditions). These conditions apply only to special Wyckoff positions and must always be added to the general conditions of the subperiodic group.

The general reflection conditions are the result of three effects: centred lattices, glide planes and screw axes. For the nine layer groups with centred lattices, the corresponding general reflection condition is [h + k = 2n]. The general reflection conditions due to glide planes and screw axes for the subperiodic groups are given in Table 1.2.13.1[link].

Table 1.2.13.1| top | pdf |
General reflection conditions due to glide planes and screw axes

(a) Layer groups.

(1) Glide planes.

Reflection conditionOrientation of planeGlide vectorSymbol
hk: h = 2n (001) a/2 a
hk: k = 2n (001) b/2 b
hk: [h+k = 2n] (001) a/2 + b/2 n
0k: k = 2n (100) b/2 b
h0: h = 2n (010) a/2 a

(2) Screw axes.

Reflection conditionDirection of axisScrew vectorSymbol
h0: h = 2n [100] a/2 21
0k: k = 2n [010] b/2 21

(b) Rod groups.

(1) Glide planes.

Reflection conditionOrientation of planeGlide vectorSymbol
l: l = 2n Any orientation parallel to the c axis c/2 c

(2) Screw axes.

Reflection conditionDirection of axisScrew vectorSymbol
l: l = 2n [001] c/2 21, 42, 63
l: l = 3n [001] c/3 31, 32, 62, 64
l: l = 4n [001] c/4 41, 43
l: l = 6n [001] c/6 61, 65

(c) Frieze groups, glide plane.

Reflection conditionOrientation of planeGlide vectorSymbol
h: h = 2n (10) a/2 g

Example: The layer group [p4bm] (L56)

General position 8d: [0k\!: k = 2n] and [h0\!: h = 2n] due respectively to the glide planes b and a. The projections along [100] and [010] of any crystal structure with this layer-group symmetry have, respectively, periodicity b/2 and a/2.

Special positions 2a and 2b: [hk\!: h + k = 2n]. Any set of equivalent atoms in either of these positions displays additional c-centring.

1.2.14. Symmetry of special projections

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1.2.14.1. Data listed in the subperiodic group tables

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Under the heading Symmetry of special projections, the following data are listed for three orthogonal projections of each layer group and rod group and two orthogonal projections of each frieze group:

  • (i) For layer and rod groups, each projection is made onto a plane normal to the projection direction. If there are three kinds of symmetry directions (cf. Table 1.2.4.1[link]), the three projection directions correspond to the primary, secondary and tertiary symmetry directions. If there are fewer than three symmetry directions, the additional projection direction(s) are taken along coordinate axes.

    For frieze groups, each projection is made on a line normal to the projection direction.

    The directions for which data are listed are as follows:

    (a) Layer groups:[\let\normalbaselines\relax\openup3pt\matrix{\left.\matrix{{\rm Triclinic/oblique}\hfill\cr {\rm Monoclinic/oblique}\hfill\cr {\rm Monoclinic/rectangular}\hfill\cr {\rm Orthorhombic/rectangular}\hfill}\right\}\hfill&[001], [100], [010] \hfill\cr &\cr\left.\matrix{{\rm Tetragonal/square}}\right.\hfill& [001], [100], [110]\hfill\cr&\cr\left.\matrix{{\rm Trigonal/hexagonal}\hfill\cr {\rm Hexagonal/hexagonal}\hfill}\right\}\hfill&[001], [100], [210]\hfill}]

    (b) Rod groups:[\let\normalbaselines\relax\openup3pt\matrix{\left.\matrix{{\rm Triclinic}\hfill\cr {\rm Monoclinic/inclined}\hfill\cr {\rm Monoclinic/orthogonal}\hfill\cr {\rm Orthorhombic}\hfill}\right\}\hfill&[001], [100], [010]\hfill\cr&\cr\left.\matrix{\rm Tetragonal}\right.\hfill&[001], [100], [110]\hfill\cr&\cr \left.\matrix{{\rm Trigonal}\hfill\cr {\rm Hexagonal}\hfill}\right\}\hfill&[001], [100], [210]\hfill}]

    (c) Frieze groups:[\let\normalbaselines\relax\openup3pt\left.\matrix{{\rm Oblique}\hfill\cr {\rm Rectangular}\hfill}\right\}\hskip8pt\quad\quad\quad\quad[10], [01]]

  • (ii) The Hermann–Mauguin symbol. For the [001] projection of a layer group, the Hermann–Mauguin symbol for the plane group resulting from the projection of the layer group is given. For the [001] projection of a rod group, the Hermann–Mauguin symbol for the resulting two-dimensional point group is given. For the remainder of the projections, in the case of both layer groups and rod groups, the Hermann–Mauguin symbol is given for the resulting frieze group. For the [10] projection of a frieze group, the Hermann–Mauguin symbol of the resulting one-dimensional point group, i.e. 1 or m, is given. For the [01] projection, the Hermann–Mauguin symbol of the resulting one-dimensional space group, i.e. p1 or pm, is given.

  • (iii) For layer groups, the basis vectors a′, b′ of the plane group resulting from the [001] projection and the basis vector a′ of the frieze groups resulting from the additional two projections are given as linear combinations of the basis vectors a, b of the layer group. Basis vectors a, b inclined to the plane of projection are replaced by the projected vectors ap, bp. For the two projections of a rod group resulting in a frieze group, the basis vector a′ of the resulting frieze group is given in terms of the basis vector c of the rod group. For the [01] projection of a frieze group, the basis vector a′ of the resulting one-dimensional space group is given in terms of the basis vector a of the frieze group.

    For rod groups and layer groups, the relations between a′, b′ and γ′ of the projected conventional basis vectors and a, b, c, α, β and γ of the conventional basis vectors of the subperiodic group are given in Table 1.2.14.1[link]. We also give in this table the relations between a′ of the projected conventional basis and a, b and γ of the conventional basis of the frieze group.

    Table 1.2.14.1| top | pdf |
    a′, b′, γ′ (a′) of the projected conventional coordinate system in terms of a, b, c, α, β, γ (a, b, γ) of the conventional coordinate system of the layer and rod groups (frieze groups)

    (a) Layer groups.

    Projection directionTriclinic/obliqueMonoclinic/oblique
    [001] [a'=a\sin\beta] [a'=a]
    [b'=b\sin \alpha] [b'=b]
    [\gamma'=180^{\circ}-\gamma^{*}] [\gamma'=\gamma]
    [100] [a'=b\sin \gamma] [a'=b\sin \gamma]
    [b'=c\sin\beta] [b'=c]
    [\gamma'=180^{\circ}-\alpha^{*}] [\gamma' = 90^{\circ}]
    [010] [a'=a\sin\gamma] [a'=a\sin\gamma]
    [b'=c\sin\alpha] [b'=c]
    [\gamma'=180^{\circ}-\beta^{*}] [\gamma'=90^{\circ}]
      Monoclinic/rectangular Orthorhombic/rectangular
    [001] [a'=a] [a'=a]
    [b'=b\sin\alpha] [b'=b]
    [\gamma'=90^{\circ}] [\gamma'=90^{\circ}]
    [100] [a'=b] [a'=b]
    [b'=c] [b'=c]
    [\gamma'=\alpha] [\gamma'=90^{\circ}]
    [010] [a'=a] [a'=a]
    [b'=c\sin\alpha] [b'=c]
    [\gamma'=90^{\circ}] [\gamma'=90^{\circ}]
      Tetragonal/square
    [001] [a'=a]
    [b'=a]
    [\gamma'=90^{\circ}]
    [100] [a'=a]
    [b'=c]
    [\gamma'=90^{\circ}]
    [110] [a'=(a/2)(2)^{1/2}]
    [b'=c]
    [\gamma'=90^{\circ}]
      Trigonal/hexagonal, hexagonal/hexagonal
    [001] [a'=a]
    [b'=a]
    [\gamma'=120^{\circ}]
    [100] [a'=[(3)^{1/2}/2]a]
    [b'=c]
    [\gamma'=90^{\circ}]
    [210] [a'=a/2]
    [b'=c]
    [\gamma'=90^{\circ}]

    (b) Rod groups.

    Projection directionTriclinicMonoclinic/inclined
    [001] [a'=a\sin\beta] [a'=a]
    [b'=b\sin\alpha] [b'=b\sin\alpha]
    [\gamma'=180^{\circ}-\gamma^{*}] [\gamma'=90^{\circ}]
    [100] [a'=c\sin\beta] [a'=c]
    [b'=b\sin\gamma] [b'=b]
    [\gamma'=180^{\circ}-\alpha^{*}] [\gamma=\alpha]
    [010] [a'=c\sin\alpha] [a'=c\sin\alpha]
    [b'=a\sin\gamma] [b'=a]
    [\gamma'=180^{\circ}-\beta^{*}] [\gamma'=90^{\circ}]
      Monoclinic/orthogonal Orthorhombic
    [001] [a'=a] [a'=a]
    [b'=b] [b'=b]
    [\gamma'=\gamma] [\gamma'=90^{\circ}]
    [100] [a'=c] [a'=c]
    [b'=b\sin\gamma] [b'=b]
    [\gamma'=90^{\circ}] [\gamma'=90^{\circ}]
    [010] [a'=c] [a'=c]
    [b'=a\sin\gamma] [b'=a]
    [\gamma'=90^{\circ}] [\gamma'=90^{\circ}]
      Tetragonal
    [001] [a'=a]
    [b'=a]
    [\gamma'=90^{\circ}]
    [100] [a'=c]
    [b'=a]
    [\gamma'=90^{\circ}]
    [110] [a'=c]
    [b'=(a/2)(2)^{1/2}]
    [\gamma'=90^{\circ}]
      Trigonal, hexagonal
    [001] [a'=a]
    [b'=a]
    [\gamma'=120^{\circ}]
    [100] [a'=c]
    [b'=[(3)^{1/2}/2]a]
    [\gamma'=90^{\circ}]
    [210] [a'=c]
    [b'=a/2]
    [\gamma'=90^{\circ}]

    (c) Frieze groups.

    Projection directionObliqueRectangular
    [10] [a'=b\sin\gamma] [a'=b]
    [01] [a'=a\sin\gamma] [a'=a]
    [\cos \alpha^{*}=(\cos\beta\cos\gamma-\cos\alpha)/(\sin\beta\sin\gamma)], [\cos \beta^{*}=(\cos\gamma\cos\alpha-\cos\beta)/(\sin\gamma\sin\alpha)], [\cos \gamma^{*}=(\cos\alpha\cos\beta-\cos\gamma)/(\sin\alpha\sin\beta).]
  • (iv) Location of the origin of the plane group, frieze group and one-dimensional space group is given with respect to the conventional lattice of the subperiodic group. The same description is used as for the location of symmetry elements (see Section 1.2.9[link]). Example: `Origin at x, 0, 0' or `Origin at x, 1/4, 0'.

1.2.14.2. Projections of centred subperiodic groups

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The only centred subperiodic groups are the nine types of centred layer groups. For the [100] and [010] projection directions, because of the centred layer-group lattice, the basis vectors of the resulting frieze groups are a′ = b/2 and a′ = a/2, respectively.

1.2.14.3. Projection of symmetry elements

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A symmetry element of a subperiodic group projects as a symmetry element only if its orientation bears a special relationship to the projection direction. In Table 1.2.14.2[link], the three-dimensional symmetry elements of the layer and rod groups and in Table 1.2.14.3[link] the two-dimensional symmetry elements of the frieze groups are listed along with the corresponding symmetry element in projection.

Table 1.2.14.2| top | pdf |
Projection of three-dimensional symmetry elements (layer and rod groups)

Symmetry element in three dimensionsSymmetry element in projection
Arbitrary orientation
Symmetry centre [\bar{1}] Rotation point 2 at projection of centre
Parallel to projection direction
Rotation axis 2, 3, 4, 6 Rotation point 2, 3, 4, 6
Screw axis 21 Rotation point 2
  31, 32   3
  41, 42, 43   4
  61, 62, 63, 64, 65   6
Rotoinversion axis [\bar{4}] Rotation point 4
  [\bar{6}\equiv 3/m]   3 (with overlap of atoms)
  [\bar{3}\equiv 3\times \bar{1}]   6
Reflection plane m Reflection line m
Glide plane with ⊥ component Glide line g
Glide plane without ⊥ component Reflection line m
Normal to projection direction
Rotation axis 2, 4, 6 Reflection line m
  3 None
Screw axis 42, 62, 64 Reflection line m
  21, 41, 43, 61, 63, 65 Glide line g
  31, 32 None
Rotoinversion axis [\bar{4}] Reflection line m parallel to axis
  [\bar{6}\equiv 3/m] Reflection line m perpendicular to axis
  [\bar{3}\equiv 3 \times \bar{1}] Rotation point 2 (at projection of centre)
Reflection plane m None, but overlap of atoms
Glide plane with glide component t Translation t
The term `with ⊥ component' refers to the component of the glide vector normal to the projection direction.

Table 1.2.14.3| top | pdf |
Projection of two-dimensional symmetry elements (frieze groups)

Symmetry element in two dimensionsSymmetry element in projection
Rotation point 2 Reflection point m
Parallel to projection direction
Reflection line m Reflection point m
Glide line g Reflection point m
Normal to projection direction
Reflection line m None (with overlap of atoms)
Glide line g with glide component t Translation t

Example: Layer group [cm2m] (L35)

Projection along [001]: This orthorhombic/rectangular plane group is centred; m perpendicular to [100] is projected as a reflection line, 2 parallel to [010] is projected as the same reflection line and m perpendicular to [001] gives rise to no symmetry element in projection, but to an overlap of atoms. Result: Plane group c1m1 (5) with a′ = a and b′ = b.

Projection along [100]: The frieze group has the basis vector a′ = b/2 due to the centred lattice of the layer group. m perpendicular to [100] gives rise only to an overlap of atoms, 2 parallel to [010] is projected as a reflection line and m perpendicular to [001] is projected as the same reflection line. Result: Frieze group [{\scr p}11m] (F4) with a′ = b/2.

Projection along [010]: The frieze group has the basis vector a′ = a/2 due to the centred lattice of the layer group. The two reflection planes project as perpendicular reflection lines and 2 parallel to [010] projects as the rotation point 2. Result: Frieze group [{\scr p}2mm] (F6) with a′ = a/2.

1.2.15. Maximal subgroups and minimal supergroups

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In IT A (2005[link]), for the representative space group of each space-group type the following information is given:

  • (i) maximal non-isomorphic subgroups,

  • (ii) maximal isomorphic subgroups of lowest index,

  • (iii) minimal non-isomorphic supergroups and

  • (iv) minimal isomorphic supergroups of lowest index.

However, Bieberbach's theorem for space groups, i.e. the classification into isomorphism classes is identical with the classification into affine equivalence classes, is not valid for subperiodic groups. Consequently, to obtain analogous tables for the subperiodic groups, we provide the following information for each representative subperiodic group:

  • (i) maximal non-isotypic non-enantiomorphic subgroups,

  • (ii) maximal isotypic subgroups and enantiomorphic subgroups of lowest index,

  • (iii) minimal non-isotypic non-enantiomorphic supergroups and

  • (iv) minimal isotypic supergroups and enantiomorphic supergroups of lowest index,

where isotypic means `belonging to the same subperiodic group type'. The cases of maximal enantiomorphic subgroups of lowest index and minimal enantiomorphic supergroups of lowest index arise only in the case of rod groups.

1.2.15.1. Maximal non-isotypic non-enantiomorphic subgroups

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The maximal non-isotypic non-enantiomorphic subgroups S of a subperiodic group G are divided into two types:

  • I translationengleiche or t subgroups and

  • II klassengleiche or k subgroups.

Type II is subdivided again into two blocks:

  • IIa: the conventional cells of G and S are the same, and

  • IIb: the conventional cell of S is larger than that of G.

Block IIa has no entries for subperiodic groups with a primitive cell. Only in the case of the nine centred layer groups are there entries, when it contains those maximal subgroups S which have lost all the centring translations of G but none of the integral translations.

1.2.15.1.1. Blocks I and IIa

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In blocks I and IIa, every maximal subgroup S of a subperiodic group G is listed with the following information:[\displaylines{\quad[i]\quad\hbox{HMS1}\quad(\hbox{HMS2})\quad\hbox{Sequence of numbers}\hfill}]The symbols have the following meaning:

  • [i]: index of S in G.

  • HMS1: short Hermann–Mauguin symbol of S, referred to the coordinate system and setting of G; this symbol may be unconventional.

  • (HMS2): conventional short Hermann–Mauguin symbol of S, given only if HMS1 is not in conventional short form.

  • Sequence of numbers: coordinate triplets of G retained in S. The numbers refer to the numbering scheme of the coordinate triplets of the general position. For the centred layer groups the following abbreviations are used:

    • Block I (all translations retained). Number +: coordinate triplet given by Number, plus that obtained by adding the centring translation (1/2, 1/2, 0) of G. (Numbers) +: the same as above, but applied to all Numbers between parentheses.

    • Block IIa (not all translations retained). Number + (1/2, 1/2, 0): coordinate triplet obtained by adding the translation (1/2, 1/2, 0) to the triplet given by Number. (Numbers) + (1/2, 1/2, 0): the same as above, but applied to all Numbers between parentheses.

Examples

  • (1) G: Layer group c211 (L10)[\matrix{{\bf I}\hfill & [2]\hfill & c1\,(p1)\hfill &1+\hfill\cr {\bf IIa}\hfill & [2]\hfill &p2_111\hfill &1\semi2+(1/2, 1/2, 0)\hfill\cr & [2]\hfill & p211\hfill & 1\semi2\hfill}]where the numbers have the following meaning:[\matrix{1+\hfill &x,y,z \quad x+1/2,y+1/2,z\hfill\cr 1\semi2\hfill &x,y,z\quad x,\bar{y}, \bar{z}\hfill\cr 1\semi 2 +\hfill &x,y,z\quad x+1/2,\bar{y}+1/2,\bar{z}\hfill \cr}]

  • (2) G: Rod group [{\scr p}422] (R30)[\matrix{{\bf I}\hfill & [2]\hfill & {\scr p}411\,({\scr p}4)\hfill & 1\semi2\semi3\semi4\cr &[2]\hfill & {\scr p}221\,({\scr p}222)\hfill & 1\semi2\semi5\semi6\hfill\cr& [2]\hfill &{\scr p}212\,({\scr p}222)\hfill& 1\semi2\semi7\semi8\hfill}]

The HMS1 symbol in each of the three subgroups S is given in the tetragonal coordinate system of the group G. In the first case, [{\scr p}411] is not the conventional short Hermann–Mauguin symbol and a second conventional symbol [{\scr p}4] is given. In the latter two cases, since the subgroups are orthorhombic rod groups, a second conventional symbol of the subgroup in an orthorhombic coordinate system is given.

1.2.15.1.2. Block IIb

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Whereas in blocks I and IIa every maximal subgroup S of G is listed, this is no longer the case for the entries of block IIb. The information given in this block is[\displaylines{\quad[i]\quad\hbox{HMS1}\quad(\hbox{Vectors})\quad(\hbox{HMS2})\hfill}]

The symbols have the following meaning:

  • [i]: index of S in G.

  • HMS1: Hermann–Mauguin symbol of S, referred to the coordinate system and setting of G; this symbol may be unconventional.

  • (Vectors): basis vectors of S in terms of the basis vectors of G. No relations are given for basis vectors which are unchanged.

  • (HMS2): conventional short Hermann–Mauguin symbol, given only if HMS1 is not in conventional short form.

Examples

  • (1) G: Rod group [{\scr p}222] (R13)[\displaylines{\quad{\bf IIb}\quad[2]\; {\scr p}222_{1}\;({\bf c}'=2{\bf c})\hfill}]There are two subgroups which obey the same basis-vector relation. Apart from the translations of the enlarged cell, the generators of the subgroups, referred to the basis vectors of the enlarged cell, are[\matrix{x,y,z\hfill&x,\bar{y},\bar{z}+1/2\hfill&\bar{x},y,\bar{z}\hfill\cr x,y,z\hfill&x,\bar{y},\bar{z}\hfill&\bar{x},y,\bar{z}+1/2.\hfill}]

  • (2) G: Layer group pm21b (L28)[\displaylines{\quad{\bf IIb}\quad[2]\; pm2_{1}n\;({\bf a}'=2{\bf a})\hfill}]This entry represents two subgroups whose generators, apart from the translations of the enlarged cell, are[\matrix{x,y,z\hfill&\bar{x}+1/2,y,z\hfill&\bar{x},y+1/2,\bar{z}\hfill\cr x,y,z\hfill&\bar{x},y,z\hfill &\bar{x}+1/2,y+1/2,\bar{z}.\hfill}]The difference between the two subgroups represented by the one entry is due to the different sets of symmetry operations of G which are retained in S. This can also be expressed as different conventional origins of S with respect to G: the two subgroups in the first example above are related by a translation c/4 of the origin, and the two subgroups in the second example by a/4.

1.2.15.2. Maximal isotypic subgroups and enantiomorphic subgroups of lowest index

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Another set of klassengleiche subgroups is that listed under IIc, i.e. the subgroups S which are of the same or of the enantiomorphic subperiodic group type as G. Again, one entry may correspond to more than one isotypic subgroup:

  • (a) As in block IIb, one entry may correspond to two isotypic subgroups whose difference can be expressed as different conventional origins of S with respect to G.

  • (b) One entry may correspond to two isotypic subgroups of equal index but with cell enlargements in different directions which are conjugate subgroups in the affine normalizer of G. The different vector relationships are given, separated by `or' and placed within one pair of parentheses; cf. example (2[link]).

Examples

  • (1) G: Rod group [{\scr p}222] (R13)[\displaylines{\quad{\bf IIc}\quad[2]\;{\scr p}222\;({\bf c}'=2{\bf c})\hfill}]This entry corresponds to two isotypic subgroups. Apart from the translations of the enlarged cell, the generators of the subgroups are[\matrix{x,y,z\hfill&x,\bar{y},\bar{z}\hfill&\bar{x},y,\bar{z}\hfill\cr x,y,z\hfill &x,\bar{y},\bar{z}+1/2\hfill&\bar{x},y,\bar{z}+1/2\hfill}]

  • (2) G: Layer group pmm2 (L23)[\displaylines{\quad{\bf IIc}\quad[2]\;pmm2\;({\bf a}'=2{\bf a}\hbox{ or }{\bf b}'=2{\bf b})\hfill}]This entry corresponds to four isotypic subgroups, two with the enlarged cell with a′ = 2a and two with the enlarged cell with b′ = 2b. The generators of these subgroups are[\matrix{{\bf a}' = 2{\bf a}\hfill&{\bf b}' = {\bf b}\hfill&x,y,z\hfill&\bar{x},y,z\hfill&x, \bar{y},z\hfill\cr {\bf a}' = 2{\bf a}\hfill&{\bf b}' = {\bf b}\hfill&x,y,z\hfill&\bar{x}+1/2,y,z\hfill&x,\bar{y},z\hfill\cr {\bf a}' = {\bf a}\hfill&{\bf b}' = 2{\bf b}\hfill&x,y,z\hfill&\bar{x},y,z\hfill&x,\bar{y},z\hfill\cr {\bf a}' = {\bf a}\hfill& {\bf b}' = 2{\bf b}\hfill& x,y,z\hfill& \bar{x},y+1/2,z\hfill& x,\bar{y},z\hfill}]

  • (3) G: Rod group [{\scr p}4_1] (R24)[\displaylines{\quad{\bf IIc}\quad[3]\,{\scr p}4_3\,({\bf c}'=3{\bf c})\hfill\cr\quad\phantom{{\bf IIc}\quad}[5]\,{\scr p}4_1\,({\bf c}'=5{\bf c})\hfill}]Listed here are both the maximal isotypic subgroup [{\scr p}4_1] and the maximal enantiomorphic subgroup [{\scr p}4_3], each of lowest index.

1.2.15.3. Minimal non-isotypic non-enantiomorphic supergroups

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If G is a maximal subgroup of a group H, then H is called a minimal supergroup of G. Minimal supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the t supergroups I of G, the listing contains the index [i] of G in H and the conventional Hermann–Mauguin symbol of H. For the k supergroups II, the subdivision between IIa and IIb is not made. The information given is similar to that for the subgroups IIb, i.e. the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of H. Note that either the conventional cell of the k supergroup H is smaller than that of the subperiodic group G, or H contains additional centring translations.

Example: G: Layer group [p2_1/m11] (L15)

Minimal non-isotypic non-enantiomorphic supergroups:[\eqalign{&{\bf I}\phantom{\bf I}\quad[2]\,pmam\semi\,[2]\,pmma\semi\,[2]\,pbma\semi\,[2]\,pmmn \cr &{\bf II}\quad[2]\,c2/m11\semi\,[2]\,p2/m11\,\,(2{\bf a}'={\bf a})}]

Block I lists [2] pmam, [2] pmma and [2] pmmn. Looking up the subgroup data of these three groups one finds [2] p21/m11. Block I also lists [2] pbma. Looking up the subgroup data of this group one finds [2] p121/m1 (p21/m11). This shows that the setting of pbma does not correspond to that of p21/m11 but rather to p121/m1. To obtain the supergroup H referred to the basis of p21/m11, the basis vectors a and b must be interchanged. This changes pbma to pmba, which is the correct symbol of the supergroup of p21/m11.

Block II contains two entries: the first where the conventional cells are the same with the supergroup having additional centring translations, and the second where the conventional cell of the supergroup is smaller than that of the original subperiodic group.

1.2.15.4. Minimal isotypic supergroups and enantiomorphic supergroups of lowest index

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No data are listed for supergroups IIc, because they can be derived directly from the corresponding data of subgroups IIc.

Example: G: Rod group [{\scr p}4_2/m] (R29)

The maximal isotypic subgroup of lowest index of [{\scr p}4_2/m] is found in block IIc: [3] [{\scr p}4_2/m] (c′ = 3c). By interchanging c′ and c, one obtains the minimal isotypic supergroup of lowest index, i.e. [3] [{\scr p}4_2/m] (3c′ = c).

1.2.16. Nomenclature

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There exists a wide variety of nomenclature for layer, rod and frieze groups (Holser, 1961[link]). Layer-group nomenclature includes zweidimensionale Raumgruppen (Alexander & Herrmann, 1929a[link],b[link]), Ebenengruppen (Weber, 1929[link]), Netzgruppen (Hermann, 1929a[link]), net groups (IT, 1952[link]; Opechowski, 1986[link]), reversal space groups in two dimensions (Cochran, 1952[link]), plane groups in three dimensions (Dornberger-Schiff, 1956[link], 1959[link]; Belov, 1959[link]), black and white space groups in two dimensions (Mackay, 1957[link]), (two-sided) plane groups (Holser, 1958[link]), Schichtgruppen (Niggli, 1959[link]; Chapuis, 1966[link]), diperiodic groups in three dimensions (Wood, 1964a[link],b[link]), layer space groups (Shubnikov & Koptsik, 1974[link]), layer groups (Köhler, 1977[link]; Koch & Fischer, 1978[link]; Vainshtein, 1981[link]; Goodman, 1984[link]; Litvin, 1989[link]), two-dimensional (subperiodic) groups in three-dimensional space (Brown et al., 1978[link]) and plane space groups in three dimensions (Grell et al., 1989[link]).

Rod-group nomenclature includes Kettengruppen (Hermann, 1929a[link],b[link]), eindimensionalen Raumgruppen (Alexander, 1929[link], 1934[link]), (crystallographic) line groups in three dimensions (IT, 1952[link]; Opechowski, 1986[link]), rod groups (Belov, 1956[link]; Vujicic et al., 1977[link]; Köhler, 1977[link]; Koch & Fischer, 1978[link]), Balkengruppen (Niggli, 1959[link]; Chapuis, 1966[link]), stem groups (Galyarskii & Zamorzaev, 1965a[link],b[link]), linear space groups (Bohm & Dornberger-Schiff, 1966[link]) and one-dimensional (subperiodic) groups in three dimensions (Brown et al., 1978[link]).

Frieze-group nomenclature includes Bortenornamente (Speiser, 1927[link]), Bandgruppen (Niggli, 1959[link]), line groups (borders) in two dimensions (IT, 1952[link]), line groups in a plane (Belov, 1956[link]), eindimensionale `zweifarbige' Gruppen (Nowacki, 1960[link]), groups of one-sided bands (Shubnikov & Koptsik, 1974[link]), ribbon groups (Köhler, 1977[link]), one-dimensional (subperiodic) groups in two-dimensional space (Brown et al., 1978[link]) and groups of borders (Vainshtein, 1981[link]).

1.2.17. Symbols

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The following general criterion was used in selecting the sets of symbols for the subperiodic groups: consistency with the symbols used for the space groups given in IT A (2005[link]). Specific criteria following from this general criterion are as follows:

  • (1) The symbols of subperiodic groups are to be of the Hermann–Mauguin (international) type. This is the type of symbol used for space groups in IT A (2005[link]).

  • (2) A symbol of a subperiodic group is to consist of a letter indicating the lattice centring type followed by a set of characters indicating symmetry elements. This is the format of the Hermann–Mauguin (international) space-group symbols in IT A (2005[link]).

  • (3) The sets of symmetry directions and their sequences in the symbols of the subperiodic groups are those of the corresponding space groups. Layer and rod groups are three-dimensional subperiodic groups of the three-dimensional space groups, and frieze groups are two-dimensional subperiodic groups of the two-dimensional space groups. Consequently, the symmetry directions and sequence of the characters indicating symmetry elements in layer and rod groups are those of the three-dimensional space groups; in frieze groups, they are those of the two-dimensional space groups, see Table 1.2.4.1[link] above and Table 2.2.4.1[link] of IT A (2005[link]). Layer groups appear as subgroups of three-dimensional space groups, as factor groups of three-dimensional reducible space groups (Kopský, 1986[link], 1988[link], 1989a[link],b[link], 1993[link]; Fuksa & Kopský, 1993[link]) and as the symmetries of planes which transect a crystal of a given three-dimensional space-group symmetry. For example, the layer group pmm2 is a subgroup of the three-dimensional space group Pmm2; is isomorphic to the factor group Pmm2/Tz of the three-dimensional space group Pmm2, where Tz is the translational subgroup of all translations along the z axis; and is the symmetry of the plane transecting a crystal of three-dimensional space-group symmetry Pmm2, perpendicular to the z axis, at z = 0. In these examples, the symbols for the three-dimensional space group and the related subperiodic layer group differ only in the letter indicating the lattice type.

A survey of sets of symbols that have been used for the subperiodic groups is given below. Considering these sets of symbols in relation to the above criteria leads to the sets of symbols for subperiodic groups used in Parts 2[link] , 3[link] and 4[link] .

1.2.17.1. Frieze groups

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A list of sets of symbols for the frieze groups is given in Table 1.2.17.1[link]. The information provided in this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 2[link] .

    Table 1.2.17.1| top | pdf |
    Frieze-group symbols

     1234567891011
    Oblique 1 [\scr p]1 r1 r1 r111 [(a)] t 1 p[1](1)1 r1 [\scr p]1
    2 [\scr p]211 r[\bar{1}'] r112 r112 [(a):2] [t:2] 5 p[2](1)1 r2 [\scr p]112
    Rectangular 3 [\scr p]1m1 r[\bar{1}] r1m rm11 [(a):m] [t:m] 3 p[1](1)m r1m [\scr p]m11
    4 [\scr p]11m r11′ rm r1m1 [(a)\cdot m] [t\cdot m] 2 p[1](m)1 r11m [\scr p]1m1
    5 [\scr p]11g r21 rg r1c1 [(a)\cdot\bar{a}] [t\cdot a] 4 p[1](c)1 r11g [\scr p]1a1
    6 [\scr p]2mm r[\bar{1}]1′ rmm2 rmm2 [(a):2\cdot m] [t:2\cdot m] 6 p[2](m)m r2mm [\scr p]mm2
    7 [\scr p]2mg r2[\bar{1}] rgm2 rmc2 [(a):2\cdot\bar{a}] [t:2\cdot a] 7 p[2](c)m r2mg [\scr p]ma2
  • Columns 3, 4 and 5: symbols listed by Opechowski (1986[link]).

  • Column 6: symbols listed by Shubnikov & Koptsik (1974[link]).

  • Column 7: symbols listed by Vainshtein (1981[link]).

  • Columns 8 and 9: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1967[link]).

  • Column 10: symbols listed by Lockwood & Macmillan (1978[link]).

  • Column 11: symbols listed by Shubnikov & Koptsik (1974[link]).

Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols of the `black and white' symmetry type (column 3) and the sets of symbols in columns 6 and 7. The sets of symbols in columns 4, 5 and 11 do not follow the sequence of symmetry directions used for two-dimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lower-case script [{\scr p}] to denote a one-dimensional lattice. The set of symbols in column 9 uses parentheses and square brackets to denote specific symmetry directions. The symbol g is used in Part 1 to denote a glide line, a standard symbol for two-dimensional space groups (IT A , 2005[link]). A letter identical with a basis-vector symbol, e.g. a or c, is not used to denote a glide line, as is done in the symbols of columns 5, 6, 7, 9 and 11, as such a letter is a standard notation for a three-dimensional glide plane (IT A , 2005[link]).

Columns 2 and 3 show the isomorphism between frieze groups and one-dimensional magnetic space groups. The one-dimensional space groups are denoted by [{\scr p}1] and [{\scr p}\bar{1}]. The list of symbols in column 3, on replacing r with [{\scr p}], is the list of one-dimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements [\bar{1}] and 1′ of the one-dimensional magnetic space groups and, respectively, the elements [m_x] and [m_y], mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.

1.2.17.2. Rod groups

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A list of sets of symbols for the rod groups is given in Table 1.2.17.2[link]. The information provided in the columns of this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 3[link] .

    Table 1.2.17.2| top | pdf |
    Rod-group symbols

     12 3456789
    Triclinic [1] [{\scr p}1] [1] [P(11)1] [1] [(a)\cdot 1] [p1] [r1] [1P1]
      [2] [{\scr p}\bar{1}] [2] [P(\bar{1}\bar{1})\bar{1}] [7] [(a)\cdot \bar{1}] [p\bar{1}] [r\bar{1}] [1P\bar{1}]
    Monoclinic/inclined [3] [{\scr p}211] [6] [P(12)1] [2] [(a):2] [p112] [r112] [1P2]
      [4] [{\scr p}m11] [3] [P(1m)1] [22] [(a)\cdot m] [p11m] [r1m1] [mP1]
      [5] [{\scr p}c11] [5] [P(1c)1] [24] [(a)\cdot \bar{a}] [p11a] [r1c1] [gP1]
      [6] [{\scr p}2/m11] [9] [P(12/m)1] [25] [(a):2:m] [p112/m] [r12/m1] [mP2]
      [7] [{\scr p}2/c11] [12] [P(12/c)1] [28] [(a):2:\bar{a}] [p112/a] [r12/c1] [gP2]
    Monoclinic/orthogonal [8] [{\scr p}112] [7] [P(11)2] [3] [(a)\cdot 2] [p211] [r211] [2P1]
      [9] [{\scr p}112_{1}] [8] [P(11)2_{1}] [8] [(a)\cdot 2_{1}] [p2_{1}] [r2_{1}] [2_{1}P1]
      [10] [{\scr p}11m] [4] [P(11)m] [23] [(a):m] [pm11] [rm11] [1Pm]
      [11] [{\scr p}112/m] [10] [P(11)2/m] [26] [(a)\cdot 2:m] [p2/m11] [r2/m11] [2Pm]
      [12] [{\scr p}112_{1}/m] [11] [P(11)2_{1}/m] [27] [(a)\cdot 2_{1}:m] [p2_{1}/m11] [r2_{1}/m11] [2_{1}Pm]
    Orthorhombic [13] [{\scr p}222] [18] [P(22)2] [61] [(a)\cdot 2:2] [p222] [r222] [2P22]
      [14] [{\scr p}222_{1}] [19] [P(22)2_{1}] [62] [(a)\cdot 2_{1}:2] [p2_{1}22] [r2_{1}22] [2_{1}P22]
      [15] [{\scr p}mm2] [13] [P(mm)2] [34] [(a)\cdot 2\cdot m] [p2mm] [r2mm] [2mmP1]
      [16] [{\scr p}cc2] [16] [P(cc)2] [35] [(a)\cdot 2\cdot \bar{a}] [p2aa] [r2cc] [2ggP1]
      [17] [{\scr p}mc2_{1}] [15] [P(mc)2_{1}] [36] [(a)\cdot 2_{1}\cdot m] [p2_{1}ma] [r2_{1}mc] [2_{1}mgP1]
      [18] [{\scr p}2mm] [14] [P(2m)m] [33] [(a):2\cdot m] [pmma] [rmm2] [mPm2]
      [19] [{\scr p}2cm] [17] [P(2c)m] [37] [(a):2\cdot \bar{a}] [pma2] [rmc2] [gPm2]
      [20] [{\scr p}mmm] [20] [P(2/m2/m)2/m] [46] [(a)\cdot m\cdot 2:m] [pmmm] [r2/m2/m2/m] [mmPm]
      [21] [{\scr p}ccm] [21] [P(2/c2/c)2/m] [47] [(a)\cdot \bar{a}\cdot 2:m] [pmaa] [r2/m2/c2/c] [ggPm]
      [22] [{\scr p}mcm] [22] [P(2/m2/c)2_{1}/m] [48] [(a)\cdot m\cdot 2_{1}:m] [pmma] [r2_{1}/m2/m2/c] [mgPm]
    Tetragonal [23] [{\scr p}4] [26] [P4(11)] [5] [(a)\cdot 4] [p4] [r4] [4P1]
      [24] [{\scr p}4_{1}] [27] [P4_{1}(11)] [11] [(a)\cdot 4_{1}] [p4_{1}] [r4_{1}] [4_{1}P1]
      [25] [{\scr p}4_{2}] [28] [P4_{2}(11)] [12] [(a)\cdot 4_{2}] [p4_{2}] [r4_{2}] [4_{2}P1]
      [26] [{\scr p}4_{3}] [29] [P4_{3}(11)] [13] [(a)\cdot 4_{3}] [p4_{3}] [r4_{3}] [4_{3}P1]
      [27] [{\scr p}\bar{4}] [23] [P\bar{4}(11)] [20] [(a)\cdot \bar{4}] [p\bar{4}] [r\bar{4}] [1P\bar{4}]
      [28] [{\scr p}4/m] [30] [P4/m(11) ] [29] [(a)\cdot 4:m] [p4/m] [r4/m] [4Pm]
      [29] [{\scr p}4_{2}/m] [31] [P4_{2}/m(11)] [30] [(a)\cdot 4_{2}:m] [p4_{2}/m] [r4_{2}/m] [4_{2}Pm]
      [30] [{\scr p}422] [35] [P4(22)] [66] [(a)\cdot 4:2] [p422] [r422] [4P22]
      [31] [{\scr p}4_{1}22] [36] [P4_{1}(22)] [67] [(a)\cdot 4_{1}:2] [p4_{1}22] [r4_{1}22] [4_{1}P22]
      [32] [{\scr p}4_{2}22] [37] [P4_{2}(22)] [68] [(a)\cdot 4_{2}:2] [p4_{2}22] [r4_{2}22] [4_{2}P22]
      [33] [{\scr p}4_{3}22] [38] [P4_{3}(22)] [69] [(a)\cdot 4_{3}:2] [p4_{3}22] [r4_{3}22] [4_{3}P22]
      [34] [{\scr p}4mm] [32] [P4(mm)] [40] [(a)\cdot 4\cdot m] [p4mm] [r4mm] [4mmP1]
      [35] [{\scr p}4_{2}cm] [33] [P4_{2}(cm)] [42] [(a)\cdot 4_{2}\cdot m] [p4_{2}ma] [r4_{2}mc] [4_{2}mgP1]
      [36] [{\scr p}4cc] [34] [P4(cc)] [41] [(a)\cdot 4\cdot \bar{a}] [p4aa] [r4cc] [4ggP1]
      [37] [{\scr p}\bar{4}2m] [24] [P\bar{4}(2m)] [49] [(a)\cdot \bar{4}\cdot m] [p\bar{4}2m] [r\bar{4}m2] [mP\bar{4}2]
      [38] [{\scr p}\bar{4}2c] [25] [P\bar{4}(2c)] [50] [(a)\cdot \bar{4}\cdot \bar{a}] [p\bar{4}2a] [r\bar{4}c2] [gP\bar{4}2]
      [39] [{\scr p}4/mmm] [39] [P4/m(2/m2/m)] [53] [(a)\cdot m\cdot 4:m] [p4/mmm] [r4/m2/m2/m] [4mmPm]
      [40] [{\scr p}4/mmc] [40] [P4/m(2/c2/c)] [54] [(a)\cdot \bar{a}\cdot 4:m] [p4/maa] [r4/m2/c2/c] [4ggPm]
      [41] [{\scr p}4_{2}/mmc] [41] [P4_{2}/m(2/m2/c)] [55] [(a)\cdot m\cdot 4_{2}:m] [p4_{2}/mma] [r4_{2}/m2/m2/c] [4_{2}mgPm]
    Trigonal [42] [{\scr p}3] [42] [P3(11)] [4] [(a)\cdot 3] [p3] [r3] [3P1]
      [43] [{\scr p}3_{1}] [43] [P3_{1}(11)] [9] [(a)\cdot 3_{1}] [p3_{1}] [r3_{1}] [3_{1}P1]
      [44] [{\scr p}3_{2}] [44] [P3_{2}(11)] [10] [(a)\cdot 3_{2}] [p3_{2}] [r3_{2}] [3_{2}P1]
      [45] [{\scr p}\bar{3}] [45] [P\bar{3}(11)] [19] [(a)\cdot \bar{6}] [p\bar{3}] [r\bar{3}] [3P\bar{1}]
      [46] [{\scr p}312] [48] [P3(21)] [63] [(a)\cdot 3:2] [p32] [r32] [3P2]
      [47] [{\scr p}3_{1}12] [49] [P3_{1}(21)] [64] [(a)\cdot 3_{1}:2] [p3_{1}2] [r3_{1}2] [3_{1}P2]
      [48] [{\scr p}3_{2}12] [50] [P3_{2}(21)] [65] [(a)\cdot 3_{2}:2] [p3_{2}2] [r3_{2}2] [3_{2}P2]
      [49] [{\scr p}3m1] [46] [P3(m1)] [38] [(a)\cdot 3\cdot m] [p3m] [r3m] [3mP1]
      [50] [{\scr p}3c1] [47] [P3(c1)] [39] [(a)\cdot 3\cdot \bar{a}] [p3a] [r3c] [3gP1]
      [51] [{\scr p}\bar{3}1m] [51] [P\bar{3}(m1)] [59] [(a)\cdot \bar{6}\cdot m] [p\bar{3}m] [r\bar{3}2/m] [3mP\bar{1}2]
      [52] [{\scr p}\bar{3}1c] [52] [P\bar{3}(c1)] [60] [(a)\cdot \bar{6}\cdot \bar{a}] [p\bar{3}a] [r\bar{3}2/c] [3gP\bar{1}2]
    Hexagonal [53] [{\scr p}6] [56] [P6(11)] [6] [(a)\cdot 6] [p6] [r6] [6P1]
      [54] [{\scr p}6_{1}] [57] [P6_{1}(11)] [14] [(a)\cdot 6_{1}] [p6_{1}] [r6_{1}] [6_{1}P1]
      [55] [{\scr p}6_{2}] [59] [P6_{2}(11)] [15] [(a)\cdot 6_{2}] [p6_{2}] [r6_{2}] [6_{2}P1]
      [56] [{\scr p}6_{3}] [61] [P6_{3}(11)] [16] [(a)\cdot 6_{3}] [p6_{3}] [r6_{3}] [6_{3}P1]
      [57] [{\scr p}6_{4}] [60] [P6_{4}(11)] [17] [(a)\cdot 6_{4}] [p6_{4}] [r6_{4}] [6_{4}P1]
      [58] [{\scr p}6_{5}] [58] [P6_{5}(11)] [18] [(a)\cdot 6_{5}] [p6_{5}] [r6_{5}] [6_{5}P1]
      [59] [{\scr p}\bar{6}] [53] [P\bar{6}(11)] [21] [(a)\cdot 3:m] [p\bar{6}] [r\bar{6}] [3Pm]
      [60] [{\scr p}6/m] [62] [P6/m(11)] [31] [(a)\cdot 6:m] [p6/m] [r6/m] [6Pm]
      [61] [{\scr p}6_{3}/m] [63] [P6_{3}/m(11)] [32] [(a)\cdot 6_{3}:m] [p6_{3}/m] [r6_{3}/m] [6_{3}Pm]
      [62] [{\scr p}622] [67] [P6(22)] [70] [(a)\cdot 6:2] [p622] [r622] [6P22]
      [63] [{\scr p}6_{1}22] [68] [P6_{1}(22)] [71] [(a)\cdot 6_{1}:2] [p6_{1}22] [r6_{1}22] [6_{1}P22]
      [64] [{\scr p}6_{2}22] [70] [P6_{2}(22)] [72] [(a)\cdot 6_{2}:2] [p6_{2}22] [r6_{2}22] [6_{2}P22]
      [65] [{\scr p}6_{3}22] [72] [P6_{3}(22)] [73] [(a)\cdot 6_{3}:2] [p6_{3}22] [r6_{3}22] [6_{3}P22]
      [66] [{\scr p}6_{4}22] [71] [P6_{4}(22)] [74] [(a)\cdot 6_{4}:2] [p6_{4}22] [r6_{4}22] [6_{4}P22]
      [67] [{\scr p}6_{5}22] [69] [P6_{5}(22)] [75] [(a)\cdot 6_{5}:2] [p6_{5}22] [r6_{5}22] [6_{5}P22]
      [68] [{\scr p}6mm] [64] [P6(mm)] [43] [(a)\cdot 6\cdot m] [p6mm] [r6mm] [6mmP1]
      [69] [{\scr p}6cc] [65] [P6(cc)] [44] [(a)\cdot 6\cdot \bar{a}] [p6aa] [r6cc] [6ggP1]
      [70] [{\scr p}6_{3}mc] [66] [P6_{3}(cm)] [45] [(a)\cdot 6_{3}\cdot m] [p6_{3}ma] [r6_{3}mc] [6_{3}mgP1]
      [71] [{\scr p}\bar{6}m2] [54] [P\bar{6}(m2)] [51] [(a)\cdot m\cdot 3:m] [p\bar{6}m2] [r\bar{6}m2] [3mPm2]
      [72] [{\scr p}\bar{6}c2] [55] [P\bar{6}(c2)] [52] [(a)\cdot \bar{a}\cdot 3:m] [p\bar{6}a2] [r\bar{6}c2] [3gPm2]
      [73] [{\scr p}6/mmm] [73] [P6/m(2/m2/m)] [56] [(a)\cdot m\cdot 6:m] [p6/mmm] [r6/m2/m2/m] [6mmPm]
      [74] [{\scr p}6/mcc] [74] [P6/m(2/c2/c)] [57] [(a)\cdot \bar{a}\cdot 6:m] [p6/maa] [r6/m2/c2/c] [6ggPm]
      [75] [{\scr p}6_{3}/mmc] [75] [P6_{3}/m(2/c2/m)] [58] [(a)\cdot m\cdot 6_{3}:m] [p6_{3}/mma] [r6_{3}/m2/m2/c] [6_{3}mgPm]
  • Columns 3 and 4: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1966[link], 1967[link]).

  • Columns 5, 6 and 7: sequential numbering and two sets of symbols listed by Shubnikov & Koptsik (1974[link]).

  • Column 8: symbols listed by Opechowski (1986[link]).

  • Column 9: symbols listed by Niggli (Chapuis, 1966[link]).

Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols in column 6 and the Niggli-type set of symbols in column 9. The set of symbols in column 8 does not use the lower-case script letter [{\scr p}], as does IT A (2005[link]), to denote a one-dimensional lattice. The order of the characters indicating symmetry elements in the set of symbols in column 7 does not follow the sequence of symmetry directions used for three-dimensional space groups. The set of symbols in column 4 have the characters indicating symmetry elements along non-lattice directions enclosed in parentheses, and do not use a lower-case script letter to denote the one-dimensional lattice. Lastly, the set of symbols in column 4, without the parentheses and with the one-dimensional lattice denoted by a lower-case script [{\scr p}], are identical with the symbols in Part 3[link] , or in some cases are the second setting of rod groups whose symbols are given in Part 3[link] . These second-setting symbols are included in the symmetry diagrams of the rod groups.

1.2.17.3. Layer groups

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A list of sets of symbols for the layer groups is given in Table 1.2.17.3[link]. The information provided in the columns of this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 4[link] .

    Table 1.2.17.3| top | pdf |
    Layer-group symbols

    (a) Columns 1–9.

     123456789
    Triclinic/oblique [1] [p1] [1] [P1] [1] [P11(1)] [1] [p1] [p1]
      [2] [p\bar{1}] [2] [P\bar{1}] [2] [P\bar{1}\bar{1}(\bar{1})] [3] [p\bar{1}] [p\bar{1}]
    Monoclinic/oblique [3] [p112] [3] [P211] [9] [P11(2)] [5] [p112] [p21]
      [4] [p11m] [4] [Pm11] [4] [P11(m)] [2] [p11m] [pm1]
      [5] [p11a] [5] [Pb11] [5] [P11(b)] [4] [p11b] [pa1]
      [6] [p112/m] [6] [P2/m11] [13] [P11(2/m)] [6] [p112/m] [p2/m1]
      [7] [p112/a] [7] [P2/b11] [17] [P11(2/b)] [7] [p112/b] [p2/a1]
    Monoclinic/rectangular [8] [p211] [8] [P112] [8] [P12(1)] [14] [p121] [p12]
      [9] [p2_{1}11] [9] [P112_{1}] [10] [P12_{1}(1)] [15] [p12_{1}1] [p12_{1}]
      [10] [c211] [10] [C112] [11] [C12(1)] [16] [c121] [c12]
      [11] [pm11] [11] [P11m] [3] [P1m(1)] [8] [p1m1] [p1m]
      [12] [pb11] [12] [P11a] [5] [P1a(1)] [10] [p1a1] [p1b]
      [13] [cm11] [13] [C11m] [7] [C1m(1)] [12] [c1m1] [c1m]
      [14] [p2/m11] [14] [P112/m] [12] [P12/m(1)] [17] [p12/m1] [p12/m]
      [15] [p2_{1}/m11] [15] [P112_{1}/m] [14] [P12_{1}/m(1)] [18] [p12_{1}/m1] [p12_{1}/m]
      [16] [p2/b11] [17] [P112/a] [16] [P12/a(1)] [20] [p12/a1] [p12/b]
      [17] [p2_{1}/b11] [18] [P112_{1}/a] [18] [P12_{1}/a(1)] [21] [p12_{1}/a1] [p12_{1}/b]
      [18] [c2/m11] [16] [C112/m] [15] [C12/m(1)] [19] [c12/m1] [c12/m]
    Orthorhombic/rectangular [19] [p222] [19] [P222] [33] [P22(2)] [37] [p222] [p222]
      [20] [p2_{1}22] [20] [P222_{1}] [34] [P2_{1}2(2)] [38] [p2_{1}22] [p222_{1}]
      [21] [p2_{1}2_{1}2] [21] [P22_{1}2_{1}] [35] [P2_{1}2_{1}(2)] [39] [p2_{1}2_{1}2] [p22_{1}2_{1}]
      [22] [c222] [22] [C222] [36] [C22(2)] [40] [c222] [c222]
      [23] [pmm2] [23] [P2mm] [19] [Pmm(2)] [22] [pmm2] [p2mm]
      [24] [pma2] [28] [P2ma] [24] [Pma(2)] [24] [pbm2] [p2ma]
      [25] [pba2] [33] [P2ba] [29] [Pba(2)] [26] [pba2] [p2ba]
      [26] [cmm2] [34] [C2mm] [30] [Cmm(2)] [28] [cmm2] [c2mm]
      [27] [pm2m] [24] [Pmm2] [20] [P2m(m)] [9] [p2mm] [pm2m]
      [28] [pm2_{1}b] [26] [Pbm2_{1}] [21] [P2_{1}m(a)] [30] [p2_{1}ma] [pa2_{1}m]
      [29] [pb2_{1}m] [25] [Pm2_{1}a] [22] [P2_{1}a(m)] [11] [p2_{1}am] [pm2_{1}a]
      [30] [pb2b] [27] [Pbb2] [23] [P2a(a)] [31] [p2aa] [pa2a]
      [31] [pm2a] [29] [Pam2] [25] [P2m(b)] [32] [p2mb] [pb2m]
      [32] [pm2_{1}n] [32] [Pnm2_{1}] [28] [P2_{1}m(n)] [35] [p2_{1}mn] [pn2_{1}m]
      [33] [pb2_{1}a] [30] [Pab2_{1}] [26] [P2_{1}a(b)] [33] [p2_{1}ab] [pb2_{1}a]
      [34] [pb2n] [31] [Pnb2] [27] [P2a(n)] [34] [p2an] [pn2a]
      [35] [cm2m] [35] [Cmm2] [31] [C2m(m)] [13] [c2mm] [cm2m]
      [36] [cm2e] [36] [Cam2] [32] [Cm2(a)] [36] [c2mb] [cb2m]
      [37] [pmmm] [37] [P2/m2/m2/m] [37] [P2/m2/m(2/m)] [23] [pmmm] [p2/m2/m2/m]
      [38] [pmaa] [38] [P2/a2/m2/a] [38] [P2/m2/a(2/a)] [41] [pmaa] [p2/a2/m2/a]
      [39] [pban] [39] [P2/n2/b2/a] [39] [P2/b2/a(2/n)] [42] [pban] [p2/n2/b2/a]
      [40] [pmam] [40] [P2/m2_{1}/m2/a] [41] [P2/b2_{1}/m(2/m)] [25] [pbmm] [p2/m2_{1}/m2/a]
      [41] [pmma] [41] [P2/a2_{1}/m2/m] [40] [P2_{1}/m2/m(2/a)] [43] [pmma] [p2/a2_{1}/m2/m]
      [42] [pman] [42] [P2/n2/m2_{1}/a] [42] [P2_{1}/b2/m(2/n)] [44] [pbmn] [p2/n2/m2_{1}/a]
      [43] [pbaa] [43] [P2/a2/b2_{1}/a] [43] [P2/b2_{1}/a(2/a)] [45] [pbaa] [p2/a2/b2_{1}/a]
      [44] [pbam] [44] [P2/m2_{1}/b2_{1}/a] [44] [P2_{1}/b2_{1}/a(2/m)] [27] [pbam] [p2/m2_{1}/b2_{1}/a]
      [45] [pbma] [45] [P2/a2_{1}/b2_{1}/m] [45] [P2_{1}/m2_{1}/a(2/b)] [46] [pmab] [p2/a2_{1}/b2_{1}/m]
      [46] [pmmn] [46] [P2/n2_{1}/m2_{1}/m] [46] [P2_{1}/m2_{1}/m(2/n)] [47] [pmmn] [p2/n2_{1}/m2_{1}/m]
      [47] [cmmm] [47] [C2/m2/m2/m] [47] [C2/m2/m(2/m)] [29] [cmmm] [c2/m2/m2/m]
      [48] [cmme] [48] [C2/a2/m2/m] [48] [C2/m2/m(2/a)] [48] [cmma] [c2/a2/m2/m]
    Tetragonal/square [49] [p4] [49] [P4] [54] [P(4)11] [50] [p4] [p4]
      [50] [p\bar{4}] [50] [P\bar{4}] [49] [P(\bar{4})11] [49] [p\bar{4}] [p\bar{4}]
      [51] [p4/m] [51] [P4/m] [55] [P(4/m)11] [51] [p4/m] [p4/m]
      [52] [p4/n] [52] [P4/n] [56] [P(4/n)11] [57] [p4/n] [p4/n]
      [53] [p422] [53] [P422] [59] [P(4)22] [55] [p422] [p422]
      [54] [p42_{1}2] [54] [P42_{1}2] [60] [P(4)2_{1}2] [56] [p42_{1}2] [p42_{1}2]
      [55] [p4mm] [55] [P4mm] [57] [P(4)mm] [52] [p4mm] [p4mm]
      [56] [p4bm] [56] [P4bm] [58] [P(4)bm] [59] [p4bm] [p4bm]
      [57] [p\bar{4}2m] [57] [P\bar{4}2m] [50] [P(\bar{4})2m] [54] [p\bar{4}2m] [p\bar{4}2m]
      [58] [p\bar{4}2_{1}m] [58] [P\bar{4}2_{1}m] [51] [P(\bar{4})2_{1}m] [60] [p\bar{4}2_{1}m] [p\bar{4}2_{1}m]
      [59] [p\bar{4}m2] [59] [P\bar{4}m2] [52] [P(\bar{4})m2] [61] [p\bar{4}m2] [p\bar{4}m2]
      [60] [p\bar{4}b2] [60] [P\bar{4}b2] [53] [P(\bar{4})b2] [64] [p\bar{4}b2] [p\bar{4}b2]
      [61] [p4/mmm] [61] [P4/m2/m2/m] [61] [P(4/m)2/m2/m] [53] [p4/mmm] [p4/m2/m2/m]
      [62] [p4/nbm] [62] [P4/n2/b2/m] [62] [P(4/n)2/b2/m] [62] [p4/nbm] [p4/n2/b2/m]
      [63] [p4/mbm] [63] [P4/m2_{1}/b2/m] [63] [P(4/m)2_{1}/b2/m] [58] [p4/mbm] [p4/m2_{1}/b2/m]
      [64] [p4/nmm] [64] [P4/n2_{1}/m2/m] [64] [P(4/n)2_{1}/m2/m] [63] [p4/nmm] [p4/n2_{1}/m2/m]
    Trigonal/hexagonal [65] [p3] [65] [P3] [65] [P(3)11] [65] [p3] [p3]
      [66] [p\bar{3}] [66] [P\bar{3}] [66] [P(\bar{3})11] [67] [p\bar{3}] [p\bar{3}]
      [67] [p312] [67] [P312] [70] [P(3)12] [72] [p312] [p312]
      [68] [p321] [68] [P321] [69] [P(3)21] [73] [p321] [p321]
      [69] [p3m1] [69] [P3m1] [67] [P(3)m1] [68] [p3m1] [p3m1]
      [70] [p31m] [70] [P31m] [68] [P(3)1m] [70] [p31m] [p31m]
      [71] [p\bar{3}1m] [71] [P\bar{3}12/m] [72] [P(\bar{3})1m] [74] [p\bar{3}1m] [p\bar{3}12/m]
      [72] [p\bar{3}m1] [72] [P\bar{3}2/m1] [71] [P(\bar{3})m1] [75] [p\bar{3}m1] [p\bar{3}2/m1]
    Hexagonal/hexagonal [73] [p6] [73] [P6] [76] [P(6)11] [76] [p6] [p6]
      [74] [p\bar{6}] [74] [P\bar{6}] [73] [P(\bar{6})11] [66] [p\bar{6}] [p\bar{6}]
      [75] [p6/m] [75] [P6/m] [77] [P(6/m)11] [77] [p6/m] [p6/m]
      [76] [p622] [76] [P622] [79] [P(6)22] [80] [p622] [p622]
      [77] [p6mm] [77] [P6mm] [78] [P(6)mm] [78] [p6mm] [p6mm]
      [78] [p\bar{6}m2] [78] [P\bar{6}m2] [74] [P(\bar{6})m2] [69] [p\bar{6}m2] [p\bar{6}m2]
      [79] [p\bar{6}2m] [79] [P\bar{6}2m] [75] [P(\bar{6})2m] [71] [p\bar{6}2m] [p\bar{6}2m]
      [80] [p6/mmm] [80] [P6/m2/m2/m] [80] [P(6/m)2/m2/m] [79] [p6/mmm] [p6/m2/m2/m]

    (b) Columns 10–17.

     11011121314151617
    Triclinic/oblique 1 [1] [C_{1}\bar{p}] [C_{1}^{1}] [1P1] [(a/b)\cdot 1] [1p1] [p1] [p1]
      2 [2] [S_{2}\bar{p}] [C_{i}^{1}] [1P\bar{1}] [(a/b)\cdot \bar{1}] [1p\bar{1}] [p2'] [p2']
    Monoclinic/oblique 3 [8] [C_{2}\bar{p}] [C_{2}^{1}] [1P2] [(a/b):2] [1p112] [p2] [p2]
      4 [3] [C_{1h}\bar{p}\mu ] [C_{1h}^{1}] [mP1] [(a/b)\cdot m] [mp1] [p^{*}1]  
      5 [4] [C_{1h}\bar{p}\alpha ] [C_{1h}^{2}] [aP1] [(a/b)\cdot \bar{b}] [bp1] [p_{b'}'1] [p_{b}'1]
      6 [12] [C_{2h}\bar{p}\mu ] [C_{2h}^{1}] [mP2] [(a/b)\cdot m:2] [mp112] [p^{*}2]  
      7 [13] [C_{2h}\bar{p}\alpha ] [C_{2h}^{2}] [aP2] [(a/b)\cdot \bar{b}:2] [bp112] [p_{b'}'2] [p_{b}'2]
    Monoclinic/rectangular 8 [9] [D_{1}\bar{p}1] [C_{2}^{2}] [1P12] [(a:b)\cdot 2] [1p12] [p1m'1] [pm']
      9 [10] [D_{1}\bar{p}2] [C_{2}^{3}] [1P12_{1}] [(a:b)\cdot 2_{1}] [1p12_{1}] [p1g'1] [pg']
      10 [11] [D_{1}\bar{c}1] [C_{2}^{4}] [1C12] [\left({{a+b}\over2}/a:b\right)\cdot 2] [1c12] [c1m'1] [cm']
      11 [5] [C_{1v}\bar{p}\mu ] [C_{1h}^{3}] [1P1m] [(a:b):m] [1p1m] [p11m] [pm]
      12 [6] [C_{1v}\bar{p}\beta ] [C_{1h}^{4}] [1P1g] [(a:b):\bar{a}] [1p1a] [p11g] [pg]
      13 [7] [C_{1v}\bar{c}\mu ] [C_{1h}^{5}] [1C1m] [\left({{a+b}\over2}/a:b\right):m] [1c1m] [c11m] [cm]
      14 [14] [D_{1d}\bar{p}\mu 1] [C_{2h}^{3}] [1P12/m] [(a:b)\cdot 2:m] [1p12/m] [p2'm'm] [pm'm]
      15 [15] [D_{1d}\bar{p}\mu 2] [C_{2h}^{5}] [1P12_{1}/m] [(a:b)\cdot 2_{1}:m] [1p12_{1}/m] [p2'g'm] [pg'm]
      16 [18] [D_{1d}\bar{p}\beta 2] [C_{2h}^{6}] [1P12/g] [(a:b)\cdot 2\cdot \bar{a}] [1p12_{1}/a] [p2'g'g] [pg'g]
      17 [17] [D_{1d}\bar{p}\beta 1] [C_{2h}^{4}] [1P12_{1}/g] [(a:b)\cdot 2_{1}:\bar{a}] [1p12/a] [p2'm'g] [pm'g]
      18 [16] [D_{1d}\bar{c}\mu 1] [C_{2h}^{7}] [1C12/m] [\left({{a+b}\over2}/a:b\right)\cdot 2:m] [1c12/m] [c2'm'm] [cm'm]
    Orthorhombic/rectangular 19 [33] [D_{2}\bar{p}11] [V^{1}] [1P222] [(a:b):2:2] [1p222] [p2m'm'] [pm'm']
      20 [34] [D_{2}\bar{p}12] [V^{3}] [1P222_{1}] [(a:b):2:2_{1}] [1p22_{1}2] [p2g'm'] [pm'g']
      21 [35] [D_{2}\bar{p}22] [V^{2}] [1P22_{1}2_{1}] [(a:b)\cdot 2_{1}:2_{1}] [1p2_{1}2_{1}2] [p2g'g'] [pg'g']
      22 [36] [D_{2}\bar{c}11] [V^{4}] [1C222] [\left({{a+b}\over2}/a:b\right):2:2] [1c222] [c2m'm'] [cm'm']
      23 [19] [C_{2v}\bar{p}\mu \mu ] [C_{2v}^{1}] [1P2mm] [(a:b):2\cdot m] [1pmm2] [p2mm] [pmm]
      24 [20] [C_{2v}\bar{p}\mu \alpha ] [C_{2v}^{2}] [1P2mg] [(a:b):2\cdot \bar{b}] [1pma2] [p2mg] [pmg]
      25 [21] [C_{2v}\bar{p}\beta \alpha ] [C_{2v}^{10}] [1P2gg] [(a:b):\bar{a}:\bar{b}] [1pba2] [p2gg] [pgg]
      26 [22] [C_{2v}\bar{c}\mu \mu ] [C_{2v}^{3}] [1C2mm] [\left({{a+b}\over2}/a:b\right):m\cdot 2] [1cmm2] [c2mm] [cmm]
      27 [23] [D_{1h}\bar{p}\mu \mu ] [C_{2v}^{4}] [mP12m] [(a:b)\cdot m\cdot 2] [mpm2] [p^{*}1m1]  
      28 [25] [D_{1h}\bar{p}\beta \mu ] [C_{2v}^{5}] [aP12_{1}m] [(a:b):m\cdot 2_{1}] [bpm2_{1}] [p_{b'}'1m1] [p_{a}'1m]
      29 [24] [D_{1h}\bar{p}\mu \beta ] [C_{2v}^{7}] [mP12_{1}g] [(a:b)\cdot m\cdot 2_{1}] [mpb2_{1}] [p^{*}1g1]  
      30 [26] [D_{1h}\bar{p}\beta \beta ] [C_{2v}^{6}] [aP12g] [(a:b)\cdot \bar{a}\cdot 2] [bpb2] [p_{b'}'1m'1] [p_{a}'1g]
      31 [27] [D_{1h}\bar{p}\alpha \mu ] [C_{2v}^{11}] [bP12m] [(a:b)\cdot \bar{b}\cdot 2] [apm2] [p_{a'}'1m1] [p_{b}'1m]
      32 [30] [D_{1h}\bar{p}\upsilon \mu ] [C_{2v}^{13}] [nP12_{1}m] [(a:b)\cdot ab\cdot 2_{1}] [npm2_{1}] [c'1m1] [p_{c}'1m]
      33 [28] [D_{1h}\bar{p}\alpha \beta ] [C_{2v}^{14}] [bP12_{1}g] [(a:b)\cdot \bar{b}:\bar{a}] [apb2_{1}] [p_{a'}'1g1] [p_{b}'1g]
      34 [29] [D_{1h}\bar{p}\upsilon \beta ] [C_{2v}^{12}] [nP12g] [(a:b)\cdot ab\cdot 2] [npb2] [c'1m'1] [p_{c}'1m']
      35 [31] [D_{1h}\bar{c}\mu \mu ] [C_{2v}^{8}] [mC12m] [\left({{a+b}\over2}/a:b\right)\cdot m\cdot 2] [mcm2] [c^{*}1m1]  
      36 [32] [D_{1h}\bar{c}\alpha \mu ] [C_{2v}^{9}] [aC12m] [\left({{a+b}\over2}/a:b\right)\cdot \bar{b}\cdot 2] [acm2] [p_{a'b'}'1m1] [c'1m]
      37 [37] [D_{2h}\bar{p}\mu \mu \mu ] [V_{h}^{1}] [mP2mm] [(a:b)\cdot m:2\cdot m] [mp2/m2/m2] [p^{*}2mm]  
      38 [38] [D_{2h}\bar{p}\alpha \mu \alpha ] [V_{h}^{5}] [aP2mg] [(a:b)\cdot \bar{a}:2\cdot \bar{a}] [ip2/m2/a2] [p_{a'}'2mg] [p_{a}'mg]
      39 [39] [D_{2h}\bar{p}\upsilon \beta \alpha ] [V_{h}^{6}] [nP2gg] [(a:b)\cdot ab:2\cdot a] [np2/b2/a2] [c'2m'm'] [p_{c}'m'm']
      40 [40] [D_{2h}\bar{p}\mu \mu \alpha ] [V_{h}^{3}] [mP2mg] [(a:b)\cdot m:2\cdot \bar{b}] [np2_{1}/m2/a2] [p^{*}2mg]  
      41 [41] [D_{2h}\bar{p}\alpha\mu\mu] [V_{h}^{9}] [aP2mm] [(a:b)\cdot \bar{a}:2\cdot m] [ap2_{1}/m2/m2] [p_{a'}'2mm] [p_{b}'mm]
      42 [42] [D_{2h}\bar{p}\upsilon \mu \alpha ] [V_{h}^{11}] [nP2mg] [(a:b)\cdot ab:2\cdot b] [np2/m2_{1}/a2] [c'2mm'] [p_{c}'m'm]
      43 [43] [D_{2h}\bar{p}\alpha \beta \alpha ] [V_{h}^{10}] [aP2gg] [(a:b)\cdot \bar{a}\cdot 2:\bar{b}] [ap2/b2_{1}/a2] [p_{a'}'2gg] [p_{b}'gg]
      44 [44] [D_{2h}\bar{p}\mu \beta \alpha ] [V_{h}^{2}] [mP2gg] [(a:b)\cdot m:\bar{a}:\bar{b}] [np2_{1}/b2_{1}/a2] [p^{*}2gg]  
      45 [45] [D_{2h}\bar{p}\alpha \beta \mu ] [V_{h}^{7}] [aP2gm] [(a:b)\cdot \bar{b}:2\cdot \bar{a}] [ap2_{1}/b2_{1}/m2] [p_{a'}'2gm] [p_{b}'mg]
      46 [46] [D_{2h}\bar{p}\upsilon \mu \mu ] [V_{h}^{8}] [nP2mm] [(a:b)\cdot ab:2\cdot m] [np2_{1}/m2_{1}/m2] [c'2mm] [p_{c}'mm]
      47 [47] [D_{2h}\bar{c}\mu \mu \mu ] [V_{h}^{4}] [mC2mm] [\left({{a+b}\over2}/a:b\right)\cdot m:2\cdot m] [mc2/m2/m2] [c^{*}2mm]  
      48 [48] [D_{2h}\bar{c}\alpha \mu \mu ] [V_{h}^{12}] [aC2mm] [\left({{a+b}\over2}/a:b\right)\cdot \bar{a}:2\cdot m] [ac2/m2/m2] [p_{a'b'}'2mm] [c'mm]
    Tetragonal/square 49 [58] [C_{4}\bar{p}] [C_{4}^{1}] [1P4] [(a:a):4] [1p4] [p4] [p4]
      50 [57] [S_{4}\bar{p}] [S_{4}^{1}] [1P\bar{4}] [(a:a):\bar{4}] [1p\bar{4}] [p4'] [p4']
      51 [61] [C_{4h}\bar{p}\mu ] [C_{4h}^{1}] [mP4] [(a:a):4:m] [mp4] [p^{*}4]  
      52 [62] [C_{4h}\bar{p}\upsilon ] [C_{4h}^{2}] [nP4] [(a:a):4:ab] [np4] [c'4] [p'4]
      53 [67] [D_{4}\bar{p}11] [D_{4}^{1}] [1P422] [(a:a):4:2] [1p422] [p4m'm'] [p4m'm']
      54 [68] [D_{4}\bar{p}21] [D_{4}^{2}] [1P42_{1}2] [(a:a):4:2_{1}] [1p42_{1}2] [p4g'm'] [p4g'm']
      55 [59] [C_{4v}\bar{p}\mu \mu ] [C_{4v}^{1}] [1P4mm] [(a:a):4\cdot m] [1p4mm] [p4mm] [p4mm]
      56 [60] [C_{4v}\bar{p}\beta \mu ] [C_{4v}^{2}] [1P4gm] [(a:a):4\odot b] [1p4bm] [p4gm] [p4gm]
      57 [63] [D_{2d}\bar{p}\mu 1] [V_{d}^{1}] [1P\bar{4}2m] [(a:a):\bar{4}:2] [1p\bar{4}2m] [p4'm'm] [p4'm'm]
      58 [64] [D_{2d}\bar{p}\mu 2] [V_{d}^{2}] [1P\bar{4}2_{1}m] [(a:a):\bar{4}\odot 2_{1}] [1p\bar{4}2_{1}m] [p4'g'm] [p4'g'm]
      59 [65] [D_{2d}\bar{c}\mu 1] [V_{d}^{3}] [1P\bar{4}m2] [(a:a):\bar{4}\cdot m] [1p\bar{4}m2] [p4'mm'] [p4'mm']
      60 [66] [D_{2d}\bar{c}\beta 1] [V_{d}^{4}] [1P\bar{4}g2] [(a:a):\bar{4}\odot \bar{b}] [1p\bar{4}b2] [p4'gm'] [p4'gm']
      61 [69] [D_{4h}\bar{p}\mu \mu \mu ] [D_{4h}^{1}] [mP4mm] [(a:a)\cdot m:4\cdot m] [mp42/m2/m] [p^{*}4mm]  
      62 [70] [D_{4h}\bar{p}\upsilon \beta \mu ] [D_{4h}^{2}] [nP4gm] [(a:a):ab:4\odot b] [np42/b2/m] [c'4m'm] [p'4gm]
      63 [71] [D_{4h}\bar{p}\mu \beta \mu ] [D_{4h}^{3}] [mP4gm] [(a:a)\cdot m:4\odot b] [mp42_{1}/b2/m] [p^{*}4gm]  
      64 [72] [D_{4h}\bar{p}\upsilon \mu \mu ] [D_{4h}^{4}] [nP4mm] [(a:a)\cdot ab:4\cdot m] [np42_{1}/m2/m] [c'4mm] [p'4mm]
    Trigonal/hexagonal 65 [49] [C_{3}\bar{c}] [C_{3}^{1}] [1P3] [(a/a):3] [1p3] [p3] [p3]
      66 [50] [S_{6}\bar{p}] [C_{3i}^{1}] [1P\bar{3}] [(a/a):\bar{3}] [1p\bar{3}] [p6'] [p6']
      67 [54] [D_{3}\bar{c}1] [D_{3}^{1}] [1P312] [(a/a):2:3] [1p312] [p3m'1] [p3m'1]
      68 [53] [D_{3}\bar{h}1] [D_{3}^{2}] [1P321] [(a/a)\cdot 2:3] [1p321] [p31m'] [p31m']
      69 [51] [C_{3v}\bar{c}\mu ] [C_{3v}^{2}] [1P3m1] [(a/a):m\cdot 3] [1p3m1] [p3m1] [p3m1]
      70 [52] [C_{3v}\bar{h}\mu ] [C_{3v}^{1}] [1P31m] [(a/a)\cdot m\cdot 3] [1p31m] [p31m] [p31m]
      71 [55] [D_{3d}\bar{c}\mu 1] [D_{3d}^{2}] [1P\bar{3}1m] [(a/a)\cdot m\cdot \bar{6}] [1p\bar{3}12/m] [p6'm'm] [p6'm'm]
      72 [56] [D_{3d}\bar{h}\mu 1] [D_{3d}^{1}] [1P\bar{3}m1] [(a/a):m\cdot \bar{6}] [1p\bar{3}2/m1] [p6'mm'] [p6'mm']
    Hexagonal/hexagonal 73 [76] [C_{6}\bar{c}] [C_{6}^{1}] [1P6] [(a/a):6] [1p6] [p6] [p6]
      74 [73] [C_{3h}\bar{c}\mu ] [C_{3h}^{1}] [mP3] [(a/a):3:m] [mp3] [p^{*}3]  
      75 [78] [C_{6h}\bar{c}\mu ] [C_{6h}^{1}] [mP6] [(a/a)\cdot m:6] [mp6] [p^{*}6]  
      76 [79] [D_{6}\bar{c}11] [D_{6}^{1}] [1P622] [(a/a)\cdot 2:6] [1p622] [p6m'm'] [p6m'm']
      77 [77] [C_{6v}\bar{c}\mu \mu ] [C_{6v}^{1}] [1P6mm] [(a/a):m\cdot 6] [1p6mm] [p6mm] [p6mm]
      78 [74] [D_{3h}\bar{c}\mu \mu ] [D_{3h}^{1}] [mP3m2] [(a/a):m\cdot 3:m] [mp3m2] [p^{*}3m1]  
      79 [75] [D_{3h}\bar{h}\mu \mu ] [D_{3h}^{2}] [mP32m] [(a/a)\cdot m:3\cdot m] [mp32m] [p^{*}31m]  
      80 [80] [D_{6h}\bar{c}\mu \mu \mu ] [D_{6h}^{1}] [mP6mm] [(a/a)\cdot m:6\cdot m] [mp6mm] [p^{*}6mm]  

    (c) Columns 18–25.

     11819202122232425
    Triclinic/oblique 1 [p1] [47]     [p1]      
      2 [p2'] [1] [p2'] [p2^{-}] [p2'] [p2[2]_{1}] [2'11] [p2/p1]
    Monoclinic/oblique 3 [p2] [48]     [p2]      
      4 [p1'] [64]     [p11']      
      5 [p_{b}'1] [2] [pt'] [pt^{-}] [p_{2b}1] [p1[2]] [b11] [p1/p1]
      6 [p21'] [65]     [p21']      
      7 [p_{b}'2] [3] [p2t'] [p2t^{-}] [p_{2b}2] [p2[2]_{2}] [2/b11] [p2/p2]
    Monoclinic/rectangular 8 [pm'] [4] [pm'] [pm^{-}] [pm'] [pm[2]_{4}] [12'1] [pm/p1]
      9 [pg'] [5] [pg'] [pg^{-}] [pg'] [pg[2]_{1}] [112_{1}'] [pg/p1]
      10 [cm'] [6 ] [cm'] [cm^{-}] [cm'] [cm[2]_{1}] [c112'] [cm/p1]
      11 [pm] [49]     [pm]      
      12 [pg] [50]     [pg]      
      13 [cm] [51 ]     [cm]      
      14 [pmm'] [14] [pmm'] [pmm^{-}] [pm'm] [pmm[2]_{2}] [2'2'2] [pmm/pm]
      15 [pmg'] [17] [pmg'] [pmg^{-}] [pmg'] [pmg[2]_{4}] [2'2_{1}'2] [pmg/pm]
      16 [pgg'] [18] [pgg'] [pgg^{-}] [pgg'] [pgg[2]_{1}] [2'2_{1}'2_{1}] [pgg/pg]
      17 [pm'g] [16] [pm'g] [pm^{-}g] [pm'g] [pmg[2]_{2}] [2'2_{1}2'] [pmg/pg]
      18 [cmm'] [21] [cmm'] [cmm^{-}] [cmm'] [cmm[2]_{2}] [c2'22'] [cmm/cm]
    Orthorhombic/rectangular 19 [pm'm'] [15] [pm'm'] [pm^{-}m^{-}] [pm'm'] [pmm[2]_{5}] [22'2'] [pmm/p2]
      20 [pm'g'] [20] [pm'g'] [pm^{-}g^{-}] [pm'g'] [pmg[2]_{5}] [22'2_{1}'] [pmg/p2]
      21 [pg'g'] [19]