# 60.2 Space Groups

A CrystGroup which has a translation subgroup of full rank is called a space group. Certain functions are available only for space groups, and not for general CrystGroups, notably all functions dealing with Wyckoff positions (see Wyckoff Positions).

Space groups which are equivalent under conjugation in the affine group (shortly: affine equivalent space groups) are said to belong to the same space group type. As is well known, in three dimensions there are 219 such space group types (if only conjugation by transformations with positive determinant is allowed, there are 230).

Representatives of all space group types in dimensions 2, 3 and 4 can be obtained from the crystallographic groups library contained in GAP (see The Crystallographic Groups Library). They must be extracted with the function `CrystGroup`, and not with the usual extraction functions `SpaceGroup` and `TransposedSpaceGroup` of that library, as these latter functions return groups which do not have an operations record that would allow to compute with them. `CrystGroup` accepts exactly the same arguments as `SpaceGroup` and `TransposedSpaceGroup`. It returns the same group as `TransposedSpaceGroup`, but equipped with a working operations record.

Space group types (and thus space groups) are classified into Z-classes and Q-classes. Two space groups belong to the same Z-class if their point groups, expressed in a basis of their respective translation lattices, are conjugate as subgroups of GL(d,Z). If the point groups are conjugate as subgoups of GL(d,Q), the two space groups are said to be in the same Q-class. This provides also a classification of point groups (expressed in a lattice basis, i.e., integral point groups) into Z-classes and Q-classes.

For a given finite integral matrix group P, representing a point group expressed in a lattice basis, a set of representative space groups for each space group type in the Z-class of P can be obtained with `SpaceGroupsPointGroup` (see SpaceGroupsPointGroup). If, moreover, the normalizer of P in GL(d,Z) is known (see NormalizerGL), exactly one representative is obtained for each space group type. Representatives of all Z-classes of maximal irreducible finite point Irreducible Maximal Finite Integral Matrix Groups) in all dimensions up to 11, and in prime dimensions up to 23. For some other dimensions, at least Q-class representatives are available. Important information about a space group is contained in its affine normalizer (see AffineNormalizer), which is the normlizer of the space group in the affine group. In a way, the affine normalizer can be regarded as the symmetry of the space group.

Warning: Groups which are called space groups in this manual should not be confused with groups extracted with `SpaceGroup` from the The Crystallographic Groups Library). The latter are not CrystGroups in the sense of this package.

GAP 3.4.4
April 1997