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. 8-14   | 1 | 2 |

## Section 1.2.6. Subperiodic group diagrams

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

### 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 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. 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).

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 general-position diagrams are indicated by the letter .

• (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.

 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. 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. 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 | top | pdf |Diagrams for monoclinic/oblique layer groups.
 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 and 1.2.6.5, 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 | top | pdf |Diagrams for monoclinic/rectangular layer groups.
 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 (). 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 () 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 of IT A (2005)].

For these groups, the two settings considered are the standard () setting and a second () setting. In Fig. 1.2.6.6, the () setting symbol is written horizontally across the top of the diagram and the second () setting symbol is written vertically on the left-hand side of the diagram. When viewing the diagram with the () 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 and 1.2.6.5). When viewing the diagram with the () 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 , 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 | 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.

 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) ()
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 (L24)

In the () setting, the Hermann–Mauguin symbol is pma2. In the () 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 and 1.2.6.8.

 Figure 1.2.6.7 | top | pdf |Diagrams for square/tetragonal layer groups.
 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 and 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, for monoclinic/inclined rod groups in Fig. 1.2.6.10, for monoclinic/orthogonal rod groups in Fig. 1.2.6.11 and for orthorhombic rod groups in Fig. 1.2.6.12.

 Figure 1.2.6.9 | top | pdf |Diagrams for triclinic rod groups.
 Figure 1.2.6.10 | top | pdf |Diagrams for monoclinic/inclined rod groups.
 Figure 1.2.6.11 | top | pdf |Diagrams for monoclinic/orthogonal rod groups.
 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. 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 , 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 (R3). The Hermann–Mauguin symbol in the conventional coordinate system in the standard () setting is given by as the translations of the rod group in the standard setting are along the direction labelled c. In the () setting, the Hermann–Mauguin symbol is , 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.

 Table 1.2.6.2| top | pdf | Distinct Hermann–Mauguin symbols for monoclinic and orthorhombic rod groups in different settings
Rod groupSetting symbol
(abc)()()(bca)()()
Hermann–Mauguin symbol
R3 c211 c121 a112 b112 b211 a121
R4 cm11 c1m1 a11m b11m bm11 a1m1
R5 cc11 c1c1 a11a b11b bb11 a1a1
R6 c2/m11 c12/m1 a112/m b112/m b2/m11 a12/m1
R7 c2/c11 c12/c1 a112/a b112/b b2/b11 a12/a1
R8 c112 c112 a211 b121 b121 a211
R9 c1121 c1121 a2111 b1211 b1211 a2111
R10 c11m c11m am11 b1m1 b1m1 am11
R11 c112/m c112/m a2/m11 b12/m1 b12/m1 a2/m11
R12 c1121/m c1121/m a21/m11 b121/m1 b121/m1 a21/m11
R13 c222 c222 a222 b222 b222 a222
R14 c2221 c2221 a2122 b2212 b2212 a2122
R15 cmm2 cmm2 a2mm bm2m bm2m a2mm
R16 ccc2 ccc2 a2aa bb2b bb2b a2aa
R17 cmc21 ccm21 a21am bb21m bm21b a21ma
R18 c2mm cm2m amm2 bmm2 b2mm am2m
R19 c2cm cc2m ama2 bbm2 b2mb am2a
R20 cmmm cmmm ammm bmmm bmmm ammm
R21 cccm cccm amaa bbmb bbmb amaa
R22 cmcm ccmm amam bbmm bmmb amma
 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 (R17)

The Hermann–Mauguin setting symbols for the six settings are:

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 and 1.2.6.15. One can consider additional settings for these rod groups: see the setting symbols in Table 1.2.6.3. 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 () setting for tetragonal groups, and in the () 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 and in the second setting where the symbol is . In Table 1.2.6.3, 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
Hermann–Mauguin symbol
R35 42cm 42mc
R37 2m m2
R38 2c c2
R41 42/mmc 42/mcm
Rod groupSetting symbol
Hermann–Mauguin symbol
R46 312 321
R47 3112 3121
R48 3212 3221
R49 3m1 31m
R50 3c1 31c
R51 1m m1
R52 1c c1
R70 63mc 63cm
R71 m2 2m
R72 c2 2c
R75 63/mmc 63/mcm
 Figure 1.2.6.14 | top | pdf |Diagrams for tetragonal rod groups.
 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 and 1.2.6.17, respectively. We consider the two settings (ab) and (), see Fig. 1.2.6.18. In the frieze-group tables, Part 2 , 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.

 Table 1.2.6.4| top | pdf | Distinct Hermann–Mauguin symbols for frieze groups in different settings
Setting symbol
(ab)()
Frieze groupHermann–Mauguin symbol
F1 a1 b1
F2 a211 b211
F3 a1m1 b11m
F4 a11m b1m1
F5 a11g b1g1
F6 a2mm b2mm
F7 a2mg b2gm
 Figure 1.2.6.16 | top | pdf |Diagrams for oblique frieze groups.
 Figure 1.2.6.17 | top | pdf |Diagrams for rectangular frieze groups.
 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.

### References

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]