GENERATORS AND RELATIONS FOR SPACE GROUPS
This section provides a minimal set of generators for each of the spacegroups, and a set of relations satisfied by them, which are sufficient to define the abstract group isomorphic to the given space group.
For each group the tabulation gives:
(i) The number assigned to the group in The International Tables for Crystallography, its Hermann-Mauguin symbol, and a list of the chosen generators;
(ii) A minimal set of relations that are sufficient to define the abstract group. A generator G of finite order n satisfies G^{n} = E. The relations of this kind are not given explicitly, they are indicated instead by the particular letter used to name the generator, according to the notational scheme given below ;
(iii) Translations expressed in terms of the chosen generators;
(iv) A particular realisation of the generators in terms of Euclidean transformations;
(v) The same Euclidean transformation specified in terms of the image of a general point [x, y, z].
Notation for translations and cyclic subgroups
E | unit element | ||
X, Y, Z | translations | t(1, 0, 0), t(0, 1, 0), t(0, 0, 1) | |
W | centring translation | t(1/2, 1/2, 0) or t(1/2, 1/2, 1/2) | |
I | inversion | I^{2} = E | |
R | diad rotation | R^{2} = E | |
M | reflection | M^{2} = E | |
F | fourfold rotation | F^{4} = E | |
B | threefold rotation | B^{3} = E | |
H | sixfold rotation or | H^{6} = E | |
Except where otherwise stated, F = 4 0, 0, z = [, x, z] and = 0, 0, z; 0, 0, 0 = [y, , ]
Except where otherwise stated, B = 3 0, 0, z = [-y, x - y, z]
For the trigonal groups H = 0, 0, z; 0, 0, 0 = [y, y - x, -z] and in the rhombohedral groups T = t(2/3, 1/3, 1/3)
For hexagonal groups H = 6 0, 0, z = [x -y, x, z] and
= 0, 0, z; 0, 0, 0 = [-x + y, -x, -z]
B = 3 x, x, x = [y, z, x]
BXB^{-1} = Z
BYB^{-1} = X
BZB^{-1} = Y