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. 94

Table 1.5.4.2 

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

Table 1.5.4.2| top | pdf |
Additional symmetry operations due to a centring vector t and their locations

Symmetry operation at the originAdditional symmetry operationsRepresentative space groups (numbers)
[C,t({1 \over 2},{1 \over 2},0)][A, t(0,{1 \over 2},{1 \over 2})][B,t({1 \over 2},0,{1 \over 2})][I,t({1 \over 2},{1 \over 2},{1 \over 2})]F
SymbolLocationSymbolLocationSymbolLocationSymbolLocationSymbolLocationSymbol
m 0, y, z b [{1 \over 4},y,z] n 0, y, z c [{1 \over 4},y,z] n [{1 \over 4},y,z] b, n, c Cmmm, Ammm, Bmmm (65)
c   n   b   m   b     Immm (71), Fmmm (69)
b   m   c   n   c     Cccm, Amaa, Bbmb (66), Ibca (73)
[d(0,{1 \over 4},{1 \over 4})]   [d(0,{3 \over 4},{1 \over 4})]   [d(0,{3 \over 4},{3 \over 4})]   [d(0,{1 \over 4},{3 \over 4})]       d, d, d Fddd (70)
m x, 0, z a [x,{1 \over 4},z] c [x,{1 \over 4},z] n x, 0, z n [x,{1 \over 4},z] a, c, n As above
a   m   n   c   c      
c   n   m   a   a      
[d({1 \over 4},0,{1 \over 4})]   [d({3 \over 4},0,{1 \over 4})]   [d({1 \over 4},0,{3 \over 4})]   [d({3 \over 4},0,{3 \over 4})]       d, d, d  
m x, y, 0 n x, y, 0 b [x,y,{1 \over 4}] a [x,y,{1 \over 4}] n [x,y,{1 \over 4}] n, b, a As above
b   a   m   n   a      
a   b   n   m   b      
[d({1 \over 4},{1 \over 4},0)]   [d({3 \over 4},{3 \over 4},0)]   [d({1 \over 4},{3 \over 4},0)]   [d({3 \over 4},{1 \over 4},0)]       d, d, d  
m x, x, z [g({1 \over 2},{1 \over 2},0)] x, x, z [g({1 \over 4},{1 \over 4},{1 \over 2})] [x,x + {1 \over 4},z] [g({1 \over 4},{1 \over 4},{1 \over 2})] [x,x - {1 \over 4},z] [n({1 \over 2},{1 \over 2},{1 \over 2})] x, x, z g, g, g I4mm (107), [F\bar{4}3m] (216)
c   [n({1 \over 2},{1 \over 2},{1 \over 2})]   [g({1 \over 4},{1 \over 4},0)]   [g({1 \over 4},{1 \over 4},0)]   [g({1 \over 2},{1 \over 2},0)]   n, g, g [F\bar{4}3c] (219)
[d({1 \over 4},{1 \over 4},{1 \over 4})]               [d({3 \over 4},{3 \over 4},{3 \over 4})]     [I\bar{4}3d] (220)
2 x, 0, 0 [2_{1}] [x,{1 \over 4},0] 2 [x,{1 \over 4},{1 \over 4}] [2_{1}] [x,0,{1 \over 4}] [2_{1}] [x,{1 \over 4},{1 \over 4}] [2_{1},2,2_{1}] C222, A222, B222 (21)
2 0, y, 0 [2_{1}] [{1 \over 4},y,0] [2_{1}] [0,y,{1 \over 4}] 2 [{1 \over 4},y,{1 \over 4}] [2_{1}] [{1 \over 4},y,{1 \over 4}] [2_{1},2_{1},2] I222 (23)
2 0, 0, z 2 [{1 \over 4},{1 \over 4},z] [2_{1}] [0,{1 \over 4},z] [2_{1}] [{1 \over 4},0,z] [2_{1}] [{1 \over 4},{1 \over 4},z] [2,2_{1},2_{1}] F222 (22)
2 [x,\bar{x},0] 2 [x,\bar{x} + {1 \over 2},0] [2_{1}(- {1 \over 4},{1 \over 4},0)] [x,\bar{x} + {1 \over 4},{1 \over 4}] [2_{1}({1 \over 4},\! - {1 \over 4},0)] [x,\bar{x} + {1 \over 4},{1 \over 4}] 2 [x,\bar{x},{1 \over 4}] [2,2_{1},2_{1}] C422 (P422) (89), I422 (97)
4 0, 0, z 4 [0,{1 \over 2},z] [4_{2}] [- {1 \over 4},{1 \over 4},z] [4_{2}] [{1 \over 4},{1 \over 4},z] [4_{2}] [0,{1 \over 2},z] [4,4_{2},4_{2}] F432 (209)
[4_{1}] 0, 0, z [4_{1}] [0,{1 \over 2},z] [4_{3}] [- {1 \over 4},{1 \over 4},z] [4_{3}] [{1 \over 4},{1 \over 4},z] [4_{3}] [0,{1 \over 2},z] [4_{1},4_{3},4_{3}] [F4_{1}32] (210)
[\bar{1}] 0, 0, 0 [\bar{1}] [{1 \over 4},{1 \over 4},0] [\bar{1}] [0,{1 \over 4},{1 \over 4}] [\bar{1}] [{1 \over 4},0,{1 \over 4}] [\bar{1}] [{1 \over 4},{1 \over 4},{1 \over 4}] [\bar{1},\bar{1},\bar{1}] Immm (71), Fmmm (69)