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
CrystGroup, and not with the usual extraction functions
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
TransposedSpaceGroup. It returns the same
TransposedSpaceGroup, but equipped with a working operations
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.
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