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. 5355

The classifications of space groups introduced in Chapter 1.3 allow one to reduce the practically unlimited number of possible space groups to a finite number of spacegroup types. However, each individual spacegroup type still consists of an infinite number of symmetry operations generated by the set of all translations of the space group. A practical way to represent the symmetry operations of space groups is based on the coset decomposition of a space group with respect to its translation subgroup, which was introduced and discussed in Section 1.3.3.2 . For our further considerations, it is important to note that the listings of the general position in the spacegroup tables can be interpreted in two ways:
With reference to a conventional coordinate system, the set of symmetry operations of a space group is described by the set of matrix–column pairs . The set of all translations forms the translation subgroup , which is a normal subgroup of of finite index [i]. If is a fixed symmetry operation, then all the products = = of translations with have the same rotation part W. Conversely, every symmetry operation of with the same matrix part W is represented in the set . The infinite set of symmetry operations is called a coset of the right coset decomposition of with respect to , and its coset representative. In this way, the symmetry operations of can be distributed into a finite set of infinite cosets, the elements of which are obtained by the combination of a coset representative and the infinite set of translations (cf. Section 1.3.3.2 ):where is omitted. Obviously, the coset representatives of the decomposition represent in a clear and compact way the infinite number of symmetry operations of the space group . Each coset in the decomposition is characterized by its linear part and its entries differ only by lattice translations. The translations form the first coset with the identity as a coset representative. The symmetry operations with rotation part form the second coset etc. The number of cosets equals the number of different matrices of the symmetry operations of the space group. This number [i] is always finite and is equal to the order of the point group of the space group (cf. Section 1.3.3.2 ).
For each space group, a set of coset representatives of the decomposition is listed under the generalposition block of the spacegroup tables. In general, any element of a coset may be chosen as a coset representative. For convenience, the representatives listed in the spacegroup tables are always chosen such that the components , of the translation parts fulfil (by subtracting integers). To save space, each matrix–column pair is represented by the corresponding coordinate triplet (cf. Section 1.2.2.3 for the shorthand notation of matrix–column pairs).
Example
The right coset decomposition of , No. 14 (unique axis b, cell choice 1) with respect to its translation subgroup is shown in Table 1.4.2.6. All possible symmetry operations of are distributed into four cosets:

The coordinate triplets of the generalposition block of (unique axis b, cell choice 1) (cf. Fig. 1.4.2.1) correspond to the coset representatives of the decomposition of the group listed in the first line of Table 1.4.2.6.
When the space group is referred to a primitive basis (which is always done for `P' space groups), each coordinate triplet of the generalposition block corresponds to one coset of , i.e. the multiplicity of the general position and the number of cosets is the same. If, however, the space group is referred to a centred cell, then the complete set of generalposition coordinate triplets is obtained by the combinations of the listed coordinate triplets with the centring translations. In this way, the total number of coordinate triplets per conventional unit cell, i.e. the multiplicity of the general position, is given by the product , where [i] is the index of in and [p] is the index of the group of integer translations in the group of all (integer and centring) translations.
Example
The listing of the general position for the space–group type Fmm2 (42) of the spacegroup tables is reproduced in Fig. 1.4.2.2. The four entries, numbered (1) to (4), are to be taken as they are printed [indicated by (0, 0, 0)+]. The additional 12 more entries are obtained by adding the centring translations to the translation parts of the printed entries [indicated by , and , respectively]. Altogether there are 16 entries, which is announced by the multiplicity of the general position, i.e. by the first number in the row. (The additional information specified on the left of the generalposition block, namely the Wyckoff letter and the site symmetry, will be dealt with in Section 1.4.4.)