International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. E, ch. 1.2, pp. 529
doi: 10.1107/97809553602060000783 Chapter 1.2. Guide to the use of the subperiodic group tables^{a}Freelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and ^{b}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA This chapter forms a guide to the entries of the subperiodic group tables given for the seven crystallographic friezegroup types (twodimensional groups with onedimensional translations) in Part 2 ; the 75 crystallographic rodgroup types (threedimensional groups with onedimensional translations) in Part 3 ; and the 80 crystallographic layergroup types (threedimensional groups with twodimensional translations) in Part 4 . 
This present volume is, in part, an extension of International Tables for Crystallography, Volume A , SpaceGroup Symmetry (IT A, 2005). Symmetry tables are given in IT A for the 230 threedimensional crystallographic spacegroup types (space groups) and the 17 twodimensional crystallographic spacegroup types (plane groups). We give in the following three parts of this volume analogous symmetry tables for the twodimensional and threedimensional subperiodic group types: the seven crystallographic friezegroup types (twodimensional groups with onedimensional translations) in Part 2 ; the 75 crystallographic rodgroup types (threedimensional groups with onedimensional translations) in Part 3 ; and the 80 crystallographic layergroup types (threedimensional groups with twodimensional translations) in Part 4 . This chapter forms a guide to the entries of the subperiodic group tables given in Parts 2–4.
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 of IT A (2005). Here we shall limit ourselves to those classifications which are explicitly used in the symmetry tables of the subperiodic groups.
The subperiodic groups are classified into affine subperiodic group types, i.e. affine equivalence classes of subperiodic groups. There are 80 affine layergroup types and seven affine friezegroup types. There are 67 crystallographic and an infinity of noncrystallographic (Opechowski, 1986) affine rodgroup types. We shall consider here only rod groups of the 67 crystallographic rodgroup types. We shall refer here to these crystallographic affine rodgroup types simply as affine rodgroup 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 rodgroup types splits into a pair of enantiomorphic crystallographic rodgroup types. Consequently, there are 75 proper affine rodgroup types. The eight pairs of enantiomorphic rodgroup types are (R24), (R26); (R31), (R33); (R43), (R44); (R47), (R48); (R54), (R58); (R55), (R57); (R63), (R67); and (R64), (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 , 3 or 4 .) We shall refer to the proper affine subperiodic group types simply as subperiodic group types.
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, and for the rod and frieze groups in the second columns of Tables 1.2.1.2 and 1.2.1.3, respectively.



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:

A subdivision of the monoclinic rodgroup category is made into monoclinic/inclined and monoclinic/orthogonal. Two different coordinate systems, see Table 1.2.1.2, 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 nonlattice basis vectors relative to the lattice vectors. For the monoclinic/inclined subdivision, the plane containing the nonlattice basis vectors is, see Fig. 1.2.1.1, inclined with respect to the lattice basis vector. For the monoclinic/orthogonal subdivision, the plane is, see Fig. 1.2.1.2, orthogonal.

Monoclinic/orthogonal basis vectors. For the monoclinic/orthogonal subdivision, α = β = 90° and the plane containing the a and b nonlattice basis vectors is orthogonal to the lattice basis vector c. 
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 threedimensional layer groups and rod groups are labelled a, b and c. The basis vectors for the twodimensional 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) and the Wyckoff positions (see Section 1.2.11). The symmetry operations are defined with respect to the directions of both lattice and nonlattice basis vectors. A Wyckoff position, denoted by a coordinate triplet (x, y, z) for the threedimensional layer and rod groups, is defined in the crystallographic coordinate system by O + r, where r = xa + yb + zc. For the twodimensional 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). In the standard setting, those basis vectors which are also lattice vectors are labelled as follows: for layer groups with their twodimensional lattice by a and b, for rod groups with their onedimensional lattice by c, and for frieze groups with their onedimensional lattice by a.
The selection of a crystallographic coordinate system is not unique. Following IT A (2005), we choose conventional crystallographic coordinate systems which have a righthanded 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 for the layer groups, and the fourth columns of Tables 1.2.1.2 and 1.2.1.3 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).
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. 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. For rod and frieze groups with their onedimensional lattices, the single cell parameter to be specified is the magnitude of the lattice basis vector.
The presentation of the subperiodic group tables in Parts 2 , 3 and 4 follows the form and content of IT A (2005). 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):
For two monoclinic/oblique layergroup 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, (i) Layer groups]. 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). These two descriptions correspond to the choice of origin, at an inversion centre and on a fourfold axis. For 15 rodgroup types, two descriptions are given, corresponding to two settings [see Section 1.2.6, (ii) Rod groups].
The description of a subperiodic group starts with a headline on a lefthand page, consisting of two or three lines which contain the following information when read from left to right.
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.

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 (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 layergroup types p1 (L1) and (L2), the rodgroup types (R1) and (R2), and the frieze group (F1).
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) and Wilson (2004).] The symbol for the Patterson subperiodic group can be deduced from the symbol of the subperiodic group in two steps:
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. The `pointgroup part' of the symbol of the Patterson symmetry represents the Laue class to which the subperiodic group belongs (cf. Tables 1.2.1.1, 1.2.1.2 and 1.2.1.3).

There are two types of diagrams, referred to as symmetry diagrams and generalposition 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. Generalposition 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 threedimensional 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 and 1.2.1.2). If the other basis vectors are not parallel to the plane of the figure, they are indicated by subscript `p', e.g. a_{p}, b_{p} and c_{p}. For frieze groups (twodimensional subperiodic groups), the diagrams are in the plane defined by the frieze group's conventional crystallographic coordinate system (see Table 1.2.1.3).
The graphical symbols for symmetry elements used in the symmetry diagrams are given in Chapter 1.1 and follow those used in IT A (2005). 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 generalposition diagrams are indicated by the letter .

The origin has been chosen according to the following conventions:

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 from centre' is given to denote the position of the origin with respect to an inversion centre.
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 threedimensional (twodimensional) space groups, because they contain threedimensional (twodimensional) translational symmetry, the asymmetric unit is a finite part of space [see Section 2.2.8 of IT A (2005)]. 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.
For all frieze groups, a limit is set on the x coordinate of the asymmetric unit by the inequalityFor the y coordinate, either there is no limit and nothing further is written, or there is the lower limit of zero, i.e. .
For all rod groups, a limit is set on the z coordinate of the asymmetric unit by the inequalityFor 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 and . Fig. 1.2.8.1 schematically shows the boundaries represented by such inequalities.

Boundaries used to define the asymmetric unit for (a) tetragonal rod groups and (b) trigonal and hexagonal rod groups. 
For all layer groups, limits are set on the x coordinate and y coordinate of the asymmetric unit by the inequalitiesFor 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 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.

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. 
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 , 8.1.5 and 11.1.1 of IT A (2005)]. 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) and the general position (Section 1.2.11). Below the geometric description we give the Seitz notation (Burns & Glazer, 1990) of each symmetry operation using the subindex notation of Zak et al. (1969).
The numbering 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. . 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.
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:

Details of this symbolism are given in Section 11.1.2 of IT A (2005).
Examples

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 . 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.
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.

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

Example: Layer group (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 ccentring 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 sitesymmetry group is the multiplicity of the general position. For position 4d, for example, the four equivalent points have the coordinates , , and . 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.
The third column of each Wyckoff position gives the site symmetry of that position. The sitesymmetry group is isomorphic to a proper or improper subgroup of the point group to which the subperiodic group under consideration belongs. Oriented sitesymmetry symbols are used to show how the symmetry elements at a site are related to the conventional crystallographic basis. The sitesymmetry symbols display the same sequence of symmetry directions as the subperiodic group symbol (cf. Table 1.2.4.1). Sets of equivalent symmetry directions that do not contribute any element to the sitesymmetry group are represented by a dot. Sets of symmetry directions having more than one equivalent direction may require more than one character if the sitesymmetry group belongs to a lower crystal system. For example, for the 2c position of tetragonal layer group p4mm (L55), the sitesymmetry 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.
The Reflection conditions are listed in the righthand column of each Wyckoff position. There are two types of reflection conditions:
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 . The general reflection conditions due to glide planes and screw axes for the subperiodic groups are given in Table 1.2.13.1.

Example: The layer group (L56)
General position 8d: and due respectively to the glide planes b and a. The projections along [100] and [010] of any crystal structure with this layergroup symmetry have, respectively, periodicity b/2 and a/2.
Special positions 2a and 2b: . Any set of equivalent atoms in either of these positions displays additional ccentring.
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:

The only centred subperiodic groups are the nine types of centred layer groups. For the [100] and [010] projection directions, because of the centred layergroup lattice, the basis vectors of the resulting frieze groups are a′ = b/2 and a′ = a/2, respectively.
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, the threedimensional symmetry elements of the layer and rod groups and in Table 1.2.14.3 the twodimensional symmetry elements of the frieze groups are listed along with the corresponding symmetry element in projection.
^{†}The term `with ⊥ component' refers to the component of the glide vector normal to the projection direction.


Example: Layer group (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 (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 (F6) with a′ = a/2.
In IT A (2005), for the representative space group of each spacegroup type the following information is given:
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:
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.
The maximal nonisotypic nonenantiomorphic subgroups S of a subperiodic group G are divided into two types:
Type II is subdivided again into two blocks:
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.
In blocks I and IIa, every maximal subgroup S of a subperiodic group G is listed with the following information:The symbols have the following meaning:

Examples
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, is not the conventional short Hermann–Mauguin symbol and a second conventional symbol 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.
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
The symbols have the following meaning:

Examples

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:

Examples

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 (L15)
Minimal nonisotypic nonenantiomorphic supergroups:
Block I lists [2] pmam, [2] pmma and [2] pmmn. Looking up the subgroup data of these three groups one finds [2] p2_{1}/m11. Block I also lists [2] pbma. Looking up the subgroup data of this group one finds [2] p12_{1}/m1 (p2_{1}/m11). This shows that the setting of pbma does not correspond to that of p2_{1}/m11 but rather to p12_{1}/m1. To obtain the supergroup H referred to the basis of p2_{1}/m11, the basis vectors a and b must be interchanged. This changes pbma to pmba, which is the correct symbol of the supergroup of p2_{1}/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.
No data are listed for supergroups IIc, because they can be derived directly from the corresponding data of subgroups IIc.
Example: G: Rod group (R29)
The maximal isotypic subgroup of lowest index of is found in block IIc: [3] (c′ = 3c). By interchanging c′ and c, one obtains the minimal isotypic supergroup of lowest index, i.e. [3] (3c′ = c).
There exists a wide variety of nomenclature for layer, rod and frieze groups (Holser, 1961). Layergroup nomenclature includes zweidimensionale Raumgruppen (Alexander & Herrmann, 1929a,b), Ebenengruppen (Weber, 1929), Netzgruppen (Hermann, 1929a), net groups (IT, 1952; Opechowski, 1986), reversal space groups in two dimensions (Cochran, 1952), plane groups in three dimensions (DornbergerSchiff, 1956, 1959; Belov, 1959), black and white space groups in two dimensions (Mackay, 1957), (twosided) plane groups (Holser, 1958), Schichtgruppen (Niggli, 1959; Chapuis, 1966), diperiodic groups in three dimensions (Wood, 1964a,b), layer space groups (Shubnikov & Koptsik, 1974), layer groups (Köhler, 1977; Koch & Fischer, 1978; Vainshtein, 1981; Goodman, 1984; Litvin, 1989), twodimensional (subperiodic) groups in threedimensional space (Brown et al., 1978) and plane space groups in three dimensions (Grell et al., 1989).
Rodgroup nomenclature includes Kettengruppen (Hermann, 1929a,b), eindimensionalen Raumgruppen (Alexander, 1929, 1934), (crystallographic) line groups in three dimensions (IT, 1952; Opechowski, 1986), rod groups (Belov, 1956; Vujicic et al., 1977; Köhler, 1977; Koch & Fischer, 1978), Balkengruppen (Niggli, 1959; Chapuis, 1966), stem groups (Galyarskii & Zamorzaev, 1965a,b), linear space groups (Bohm & DornbergerSchiff, 1966) and onedimensional (subperiodic) groups in three dimensions (Brown et al., 1978).
Friezegroup nomenclature includes Bortenornamente (Speiser, 1927), Bandgruppen (Niggli, 1959), line groups (borders) in two dimensions (IT, 1952), line groups in a plane (Belov, 1956), eindimensionale `zweifarbige' Gruppen (Nowacki, 1960), groups of onesided bands (Shubnikov & Koptsik, 1974), ribbon groups (Köhler, 1977), onedimensional (subperiodic) groups in twodimensional space (Brown et al., 1978) and groups of borders (Vainshtein, 1981).
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). Specific criteria following from this general criterion are as follows:

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 , 3 and 4 .
A list of sets of symbols for the frieze groups is given in Table 1.2.17.1. The information provided in this table is as follows:

Sets of symbols which are of a nonHermann–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 twodimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lowercase script to denote a onedimensional 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 twodimensional space groups (IT A , 2005). A letter identical with a basisvector 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 threedimensional glide plane (IT A , 2005).
Columns 2 and 3 show the isomorphism between frieze groups and onedimensional magnetic space groups. The onedimensional space groups are denoted by and . The list of symbols in column 3, on replacing r with , is the list of onedimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements and 1′ of the onedimensional magnetic space groups and, respectively, the elements and , mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.
A list of sets of symbols for the rod groups is given in Table 1.2.17.2. The information provided in the columns of this table is as follows:

Sets of symbols which are of a nonHermann–Mauguin (international) type are the set of symbols in column 6 and the Nigglitype set of symbols in column 9. The set of symbols in column 8 does not use the lowercase script letter , as does IT A (2005), to denote a onedimensional 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 threedimensional space groups. The set of symbols in column 4 have the characters indicating symmetry elements along nonlattice directions enclosed in parentheses, and do not use a lowercase script letter to denote the onedimensional lattice. Lastly, the set of symbols in column 4, without the parentheses and with the onedimensional lattice denoted by a lowercase script , are identical with the symbols in Part 3 , or in some cases are the second setting of rod groups whose symbols are given in Part 3 . These secondsetting symbols are included in the symmetry diagrams of the rod groups.
A list of sets of symbols for the layer groups is given in Table 1.2.17.3. The information provided in the columns of this table is as follows:
