(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 generalposition diagram on the right and the symmetry diagram on the left. These diagrams are illustrated in Fig. 1.2.6.1.
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 1.2.6.3
 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.
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 lefthand 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 lefthand 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 lefthand 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 lefthand side.

Figure 1.2.6.6
 top  pdf  Monoclinic/rectangular and orthorhombic/rectangular layer groups with two settings. For the secondsetting symbol printed vertically, the page must be turned clockwise by 90° or viewed from the righthand 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.
Layer group  Setting symbol 
(abc)  () 
Hermann–Mauguin symbol 
L8 
p211 
p121 
L9 
p2_{1}11 
p12_{1}1 
L10 
c211 
c121 
L11 
pm11 
p1m1 
L12 
pb11 
p1a1 
L13 
cm11 
c1m1 
L14 
p2/m11 
p12/m1 
L15 
p2_{1}/m11 
p12_{1}/m1 
L16 
p2/b11 
p12/a1 
L17 
p2_{1}/b11 
p12_{1}/a1 
L18 
c2/m11 
c12/m1 
L20 
p2_{1}22 
p22_{1}2 
L24 
pma2 
pbm2 
L27 
pm2m 
p2mm 
L28 
pm2_{1}b 
p2_{1}ma 
L29 
pb2_{1}m 
p2_{1}am 
L30 
pb2b 
p2aa 
L31 
pm2a 
p2mb 
L32 
pm2_{1}n 
p2_{1}mn 
L33 
pb2_{1}a 
p2_{1}ab 
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 

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.
(ii) Rod groups
For triclinic, monoclinic/inclined, monoclinic/orthogonal and orthorhombic rod groups, six diagrams are given: three symmetry diagrams and three generalposition 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 nonlattice 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.
The symmetry diagrams consist of the c projection, outlined with a circle at the upper lefthand side, the a projection at the lower lefthand side and the b projection at the upper righthand side. The generalposition diagrams are the c projection, outlined with a circle at the lower righthand side, and the remaining two generalposition 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 lefthand 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.
Rod group  Setting symbol 
(abc)  ()  ()  (bca)  ()  () 
Hermann–Mauguin symbol 
R3 
_{c}211 
_{c}121 
_{a}112 
_{b}112 
_{b}211 
_{a}121 
R4 
_{c}m11 
_{c}1m1 
_{a}11m 
_{b}11m 
_{b}m11 
_{a}1m1 
R5 
_{c}c11 
_{c}1c1 
_{a}11a 
_{b}11b 
_{b}b11 
_{a}1a1 
R6 
_{c}2/m11 
_{c}12/m1 
_{a}112/m 
_{b}112/m 
_{b}2/m11 
_{a}12/m1 
R7 
_{c}2/c11 
_{c}12/c1 
_{a}112/a 
_{b}112/b 
_{b}2/b11 
_{a}12/a1 
R8 
_{c}112 
_{c}112 
_{a}211 
_{b}121 
_{b}121 
_{a}211 
R9 
_{c}112_{1} 
_{c}112_{1} 
_{a}2_{1}11 
_{b}12_{1}1 
_{b}12_{1}1 
_{a}2_{1}11 
R10 
_{c}11m 
_{c}11m 
_{a}m11 
_{b}1m1 
_{b}1m1 
_{a}m11 
R11 
_{c}112/m 
_{c}112/m 
_{a}2/m11 
_{b}12/m1 
_{b}12/m1 
_{a}2/m11 
R12 
_{c}112_{1}/m 
_{c}112_{1}/m 
_{a}2_{1}/m11 
_{b}12_{1}/m1 
_{b}12_{1}/m1 
_{a}2_{1}/m11 
R13 
_{c}222 
_{c}222 
_{a}222 
_{b}222 
_{b}222 
_{a}222 
R14 
_{c}222_{1} 
_{c}222_{1} 
_{a}2_{1}22 
_{b}22_{1}2 
_{b}22_{1}2 
_{a}2_{1}22 
R15 
_{c}mm2 
_{c}mm2 
_{a}2mm 
_{b}m2m 
_{b}m2m 
_{a}2mm 
R16 
_{c}cc2 
_{c}cc2 
_{a}2aa 
_{b}b2b 
_{b}b2b 
_{a}2aa 
R17 
_{c}mc2_{1} 
_{c}cm2_{1} 
_{a}2_{1}am 
_{b}b2_{1}m 
_{b}m2_{1}b 
_{a}2_{1}ma 
R18 
_{c}2mm 
_{c}m2m 
_{a}mm2 
_{b}mm2 
_{b}2mm 
_{a}m2m 
R19 
_{c}2cm 
_{c}c2m 
_{a}ma2 
_{b}bm2 
_{b}2mb 
_{a}m2a 
R20 
_{c}mmm 
_{c}mmm 
_{a}mmm 
_{b}mmm 
_{b}mmm 
_{a}mmm 
R21 
_{c}ccm 
_{c}ccm 
_{a}maa 
_{b}bmb 
_{b}bmb 
_{a}maa 
R22 
_{c}mcm 
_{c}cmm 
_{a}mam 
_{b}bmm 
_{b}mmb 
_{a}mma 


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 generalposition 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.
Rod group  Setting symbol 
 
Hermann–Mauguin symbol 
R35 
4_{2}cm 
4_{2}mc 
R37 
2m 
m2 
R38 
2c 
c2 
R41 
4_{2}/mmc 
4_{2}/mcm 
Rod group  Setting symbol 
 
Hermann–Mauguin symbol 
R46 
312 
321 
R47 
3_{1}12 
3_{1}21 
R48 
3_{2}12 
3_{2}21 
R49 
3m1 
31m 
R50 
3c1 
31c 
R51 
1m 
m1 
R52 
1c 
c1 
R70 
6_{3}mc 
6_{3}cm 
R71 
m2 
2m 
R72 
c2 
2c 
R75 
6_{3}/mmc 
6_{3}/mcm 

(iii) Frieze groups
Two diagrams are given for each frieze group: a symmetry diagram and a generalposition 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 friezegroup 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.
 Setting symbol 
 (ab)  () 
Frieze group  Hermann–Mauguin symbol 
F1 
_{a}1 
_{b}1 
F2 
_{a}211 
_{b}211 
F3 
_{a}1m1 
_{b}11m 
F4 
_{a}11m 
_{b}1m1 
F5 
_{a}11g 
_{b}1g1 
F6 
_{a}2mm 
_{b}2mm 
F7 
_{a}2mg 
_{b}2gm 


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 righthand side.

