International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 2931
Section 1.3.3.2. Coset decomposition with respect to the translation subgroup^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands 
The translation subgroup of a space group can be used to distribute the operations of into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of into cosets with respect to (see Section 1.1.4 for details of cosets).
Definition
Let be a space group with translation subgroup .
If the translation subgroup is a subgroup of index [i] in , a set of coset representatives for relative to consists of [i] operations , where is assumed to be the identity element of . The cosets of relative to can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table 1.3.3.3.

Remark: We can assume some enumeration of the operations in because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in are simply columns with integral components . A straightforward enumeration of these columns would start with
Writing out the matrix–column pairs, the coset consists of the operations of the form with running over the lattice translations of . This means that the operations of a coset with respect to the translation subgroup all have the same linear part, which is also evident from a listing of the cosets as columns of an infinite array, as in the example above.
The onetoone correspondence between the pointgroup operations and the cosets relative to explicitly displays the isomorphism between the point group of and the factor group . This correspondence is also exploited in the listing of the generalposition coordinates. What is given there are the coordinate triplets for coset representatives of relative to , which correspond to the first row of the array in Table 1.3.3.3. As just explained, the other operations in can be obtained from these coset representatives by adding a lattice translation to the translational part.
Furthermore, the correspondence between the point group and the coset decomposition relative to makes it easy to find a system of coset representatives of relative to . What is required is that the linear parts of the are precisely the operations in the point group of . If are the different operations in the point group of , then a system of coset representatives is obtained by choosing for every linear part a translation part such that is an operation in .
It is customary to choose the translation parts of the coset representatives such that their coordinates lie between 0 and 1, excluding 1. In particular, if the translation part of a coset representative is a lattice vector, it is usually chosen as the zero vector .
Note that due to the fact that is a normal subgroup of , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.