International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.5, p. 93

Table 1.5.4.1 

B. Souvignier,c G. Chapuisd and H. Wondratscheka

Table 1.5.4.1| top | pdf |
Additional symmetry operations and their locations if the translation vector t is inclined to the symmetry axis or symmetry plane

The table is restricted to integral translations and thus is valid for primitive lattices and for integral translations in centred lattices (for centring translations see Table 1.5.4.2[link]).

Symmetry operation at the originTranslation vector tAdditional symmetry operationRepresentative plane and space groups (numbers)
SymbolLocationSymbolScrew or glide componentLocation
Tetragonal, rhombohedral and cubic coordinate systems
2 x, x, 0 1, 0, 0 [2_{1}] [{1 \over 2},{1 \over 2},0] [x,x + {1 \over 2},0] P422 (89)
    0, 1, 0   [{1 \over 2},{1 \over 2},0]   R32 (155)
            P432 (207)
m x, x, z 1, 0, 0 g [{1 \over 2},{1 \over 2},0] [x,x + {1 \over 2},z] p4mm (11)
    0, 1, 0   [{1 \over 2},{1 \over 2},0]   P4mm (99)
            R3m (160)
            [P\bar{4}3m] (215)
c x, x, z 1, 0, 0 n [{1 \over 2},{1 \over 2},{1 \over 2}] [x,x + {1 \over 2},z] [P\bar{4}2c] (112)
    0, 1, 0   [{1 \over 2},{1 \over 2},{1 \over 2}]   R3c (161)
            [P\bar{4}3n] (218)
Hexagonal coordinate system
2 x, 0, 0 1, 1, 0 [2_{1}] [{1 \over 2},0,0] [x,{1 \over 2},0] P321 (150)
    0, 1, 0   [-{1 \over 2},0,0]   R32 (155)
2 x, 2x, 0 0, 1, 0 [2_{1}] [{1 \over 2},1,0] [x,2x + {1 \over 2},0] P312 (149)
    1, 1, 0       P622 (177)
m x, 2x, z 0, 1, 0 b [{1 \over 2},1,0] [x,2x + {1 \over 2},z] P3m1 (156)
    1, 1, 0       p3m1 (14)
            R3m (160)
c x, 2x, z 0, 1, 0 n [{1 \over 2},1,{1 \over 2}] [x,2x + {1 \over 2},z] P3c1 (158)
    1, 1, 0       [P\bar{6}c2] (188)
            R3c (161)
m x, 0, z 1, 1, 0 a [{1 \over 2},0,0] [x,{1 \over 2},z] P31m (157)
    0, 1, 0   [-{1 \over 2},0,0]   p31m (15)
c x, 0, z 1, 1, 0 n [{1 \over 2},0,{1 \over 2}] [x,{1 \over 2},z] P31c (159)
    0, 1, 0   [-{1 \over 2},0,{1 \over 2}]   [P\bar{6}2c] (190)
Rhombohedral and cubic coordinate systems
3 x, x, x 1, 0, 0 [3_{1}] [{1 \over 3},{1 \over 3},{1 \over 3}] [x,x + {2 \over 3},x + {1 \over 3}] R3 (146)
0, 1, 0
0, 0, 1
3 x, x, x 2, 0, 0 [3_{2}] [{2 \over 3},{2 \over 3},{2 \over 3}] [x,x + {1 \over 3},x + {2 \over 3}] P23 (195)
0, 2, 0
0, 0, 2