International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 59-61

## Section 1.4.3. Generation of space groups

H. Wondratscheke

### 1.4.3. Generation of space groups

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In group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as a finite ordered product of the generators. For space groups of one, two and three dimensions, generators may always be chosen and ordered in such a way that each symmetry operation can be written as the product of powers of h generators (j = ). Thus,where the powers are positive or negative integers (including zero). The description of a group by means of generators has the advantage of compactness. For instance, the 48 symmetry operations in point group can be described by two generators. Different choices of generators are possible. For the space-group tables, generators and generating procedures have been chosen such as to make the entries in the blocks General position' (cf. Section 2.1.3.11 ) and Symmetry operations' (cf. Section 2.1.3.9 ) as transparent as possible. Space groups of the same crystal class are generated in the same way (see Table 1.4.3.1 for the sequences that have been chosen), and the aim has been to accentuate important subgroups of space groups as much as possible. Accordingly, a process of generation in the form of a composition series has been adopted, see Ledermann (1976). The generator is defined as the identity operation, represented by (1) . The generators , , and are the translations with translation vectors a, b and c, respectively. Thus, the coefficients , and may have any integral value. If centring translations exist, they are generated by translations (and in the case of an F lattice) with translation vectors d (and e). For a C lattice, for example, d is given by . The exponents (and ) are restricted to the following values:

• Lattice letter A, B, C, I: or 1.

 Table 1.4.3.1| top | pdf | Sequence of generators for the crystal classes
 The space-group generators differ from those listed here by their glide or screw components. The generator 1 is omitted, except for crystal class 1. The generators are represented by the corresponding Seitz symbols (cf. Tables 1.4.2.1 –1.4.2.3 ). Following the conventions, the subscript of a symbol denotes the characteristic direction of that operation, where necessary. For example, the subscripts 001, 010, 110 etc. refer to the directions [001], [010], [110] etc. For mirror reflections m, the `direction of m' refers to the normal of the mirror plane.
Hermann–Mauguin symbol of crystal classGenerators (sequence left to right)
1 1
2 2
m m
222
mm2
mmm
4
422
4mm
3
(rhombohedral coordinates )
(rhombohedral coordinates
321
(rhombohedral coordinates )
312
3m1
(rhombohedral coordinates )
31m
(rhombohedral coordinates )
6
622
6mm
23
432
• Lattice letter R (hexagonal axes): , 1 or 2.

• Lattice letter : or 1; or 1.

As a consequence, any translation of with translation vectorcan be obtained as a productwhere are integers determined by . The generators and are enclosed between parentheses because they are effective only in centred lattices.

The remaining generators generate those symmetry operations that are not translations. They are chosen in such a way that only terms or occur. For further specific rules, see below.

The process of generating the entries of the space-group tables may be demonstrated by the example in Table 1.4.3.2, where denotes the group generated by . For , the next generator is introduced when , because in this case no new symmetry operation would be generated by . The generating process is terminated when there is no further generator. In the present example, completes the generation: (178).

 Table 1.4.3.2| top | pdf | Generation of the space group (178)
 The entries in the second column designated by the numbers (1)–(12) correspond to the coordinate triplets of the general position of .
Coordinate tripletsSymmetry operations
(1) ; Identity
The group of all translations of P6122 has been generated
Threefold screw rotation
Threefold screw rotation
Now the space group has been generated
Twofold screw rotation
Sixfold screw rotation
Sixfold screw rotation
Now the space group has been generated
Twofold rotation, direction of axis [110]
Twofold rotation, axis [100]
Twofold rotation, axis [010]
Twofold rotation, axis
Twofold rotation, axis [120]
Twofold rotation, axis [210]

#### 1.4.3.1. Selected order for non-translational generators

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For the non-translational generators, the following sequence has been adopted:

• (a) In all centrosymmetric space groups, an inversion (if possible at the origin O) has been selected as the last generator.

• (b) Rotations precede symmetry operations of the second kind. In crystal classes and and and , as an exception, and are generated first in order to take into account the conventional choice of origin in the fixed points of and .

• (c) The non-translational generators of space groups with C, A, B, F, I or R symbols are those of the corresponding space group with a P symbol, if possible. For instance, the generators of (24) are those of (19) and the generators of Ibca (73) are those of Pbca (61), apart from the centring translations.

Exceptions: I4cm (108) and I4/mcm (140) are generated via P4cc (103) and P4/mcc (124), because P4cm and P4/mcm do not exist. In space groups with d glides (except , No. 122) and also in (88), the corresponding rotation subgroup has been generated first. The generators of this subgroup are the same as those of the corresponding space group with a lattice symbol P.

#### Example

(227):

.

• (d) In some cases, rule (c) could not be followed without breaking rule (a), e.g. in Cmme (67). In such cases, the generators are chosen to correspond to the Hermann–Mauguin symbol as far as possible. For instance, the generators (apart from centring) of Cmme and Imma (74) are those of Pmmb, which is a non-standard setting of Pmma (51). (A combination of the generators of Pmma with the C- or I-centring translation results in non-standard settings of Cmme and Imma.)

For the space groups with lattice symbol P, the generation procedure has given the same triplets (except for their sequence) as in IT (1952). In non-P space groups, the triplets listed sometimes differ from those of IT (1952) by a centring translation.

### References

International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
Ledermann, W. (1976). Introduction to Group Theory. London: Longman.