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 Gn = 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].


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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 I2 = E
R diad rotation R2 = E
M reflection M2 = E
F fourfold rotation F4 = E
B threefold rotation B3 = E      
H sixfold rotation or H6 = E

Triclinic, Monoclinic & Orthorhombic


Except where otherwise stated, F = 4  0, 0, z = [, x, z] and =   0, 0, z; 0, 0, 0 = [y, , ]

Trigonal & Hexagonal

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