International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 5961
Section 1.4.3. Generation of space groups
H. Wondratschek^{e}

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 spacegroup 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:

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 spacegroup 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).

For the nontranslational generators, the following sequence has been adopted:
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 nonP space groups, the triplets listed sometimes differ from those of IT (1952) by a centring translation.
References
International Tables for Xray 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.