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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 52

Table 1.4.2.1 

M. I. Aroyo,a G. Chapuis,b B. Souvignierd and A. M. Glazerc

Table 1.4.2.1| top | pdf |
Linear parts R of the Seitz symbols [\{{\bi R}|{\bi v}\}] for space-group symmetry operations of cubic, tetragonal, orthorhombic, monoclinic and triclinic crystal systems

Each symmetry operation is specified by the shorthand description of the rotation part of its matrix–column presentation, the type of symmetry operation and its characteristic direction.

IT A descriptionSeitz symbol
No.Coordinate tripletTypeOrientation
1 [x,y,z] 1   1
2 [\bar x,\bar y,z] 2 [0,0,z] [{2_{001}}]
3 [\bar x,y,\bar z] 2 [0,y,0] [{2_{010}}]
4 [x,\bar y,\bar z] 2 [x,0,0] [{2_{100}}]
5 [z,x,y] [{3^ + }] [x,x,x] [3_{{111}}^ +]
6 [z,\bar x,\bar y] [{3^ + }] [\bar x,x,\bar x] [3_{{\bar 11\bar 1}}^ +]
7 [\bar z,\bar x,y] [{3^ + }] [x,\bar x,\bar x] [3_{{1\bar 1\bar 1}}^ +]
8 [\bar z,x,\bar y] [{3^ + }] [\bar x,\bar x,x] [3_{{\bar 1\bar 11}}^ +]
9 [y,z,x] [{3^ - }] [x,x,x] [3_{{111}}^ -]
10 [\bar y,z,\bar x] [{3^ - }] [x,\bar x,\bar x] [3_{{1\bar 1\bar 1}}^ -]
11 [y,\bar z,\bar x] [{3^ - }] [\bar x,\bar x,x] [3_{{\bar 1\bar 11}}^ -]
12 [\bar y,\bar z,x] [{3^ - }] [\bar x,x,\bar x] [3_{{\bar 11\bar 1}}^ -]
13 [y,x,\bar z] 2 [x,x,0] [{2_{110}}]
14 [\bar y,\bar x,\bar z] 2 [x,\bar x,0] [{2_{1\bar 10}}]
15 [y,\bar x,z] [{4^ - }] [0,0,z] [4_{{001}}^ -]
16 [\bar y,x,z] [{4^ + }] [0,0,z] [4_{{001}}^ +]
17 [x,z,\bar y] [{4^ - }] [x,0,0] [4_{{100}}^ -]
18 [\bar x,z,y] 2 [0,y,y] [{2_{011}}]
19 [\bar x,\bar z,\bar y] 2 [0,y,\bar y] [{2_{01\bar 1}}]
20 [x,\bar z,y] [{4^ + }] [x,0,0] [4_{{100}}^ +]
21 [z,y,\bar x] [{4^ + }] [0,y,0] [4_{{010}}^ +]
22 [z,\bar y,x] 2 [x,0,x] [{2_{101}}]
23 [\bar z,y,x] [{4^ - }] [0,y,0] [4_{{010}}^ -]
24 [\bar z,\bar y,\bar x] 2 [\bar x,0,x] [{2_{\bar 101}}]
25 [\bar x,\bar y,\bar z] [\bar 1]   [\bar 1]
26 [x,y,\bar z] m [x,y,0] [{m_{001}}]
27 [x,\bar y,z] m [x,0,z] [{m_{010}}]
28 [\bar x,y,z] m [0,y,z] [{m_{100}}]
29 [\bar z,\bar x,\bar y] [{\bar 3^ + }] [x,x,x] [\bar 3_{111}^ +]
30 [\bar z,x,y] [{\bar 3^ + }] [\bar x,x,\bar x] [\bar 3_{{\bar 11\bar 1}}^ +]
31 [z,x,\bar y] [{\bar 3^ + }] [x,\bar x,\bar x] [\bar 3_{{1\bar 1\bar 1}}^ +]
32 [z,\bar x,y] [{\bar 3^ + }] [\bar x,\bar x,x] [3_{{\bar 1\bar 11}}^ +]
33 [\bar y,\bar z,\bar x] [{\bar 3^ - }] [x,x,x] [\bar 3_{111}^ -]
34 [y,\bar z,x] [{\bar 3^ - }] [x,\bar x,\bar x] [\bar 3_{{1\bar 1\bar 1}}^ -]
35 [\bar y,z,x] [{\bar 3^ - }] [\bar x,\bar x,x] [\bar 3_{{\bar 1\bar 11}}^ -]
36 [y,z,\bar x] [{\bar 3^ - }] [\bar x,x,\bar x] [\bar 3_{{\bar 11\bar 1}}^ -]
37 [\bar y,\bar x,z] m [x,\bar x,z] [{m_{110}}]
38 [y,x,z] m [x,x,z] [{m_{1\bar 10}}]
39 [\bar y,x,\bar z] [{\bar 4^ - }] [0,0,z] [\bar 4_{{001}}^ -]
40 [y,\bar x,\bar z] [{\bar 4^ + }] [0,0,z] [\bar 4_{{001}}^ +]
41 [\bar x,\bar z,y] [{\bar 4^ - }] [x,0,0] [\bar 4_{{100}}^ -]
42 [x,\bar z,\bar y] m [x,y,\bar y] [{m_{011}}]
43 [x,z,y] m [x,y,y] [{m_{01\bar 1}}]
44 [\bar x,z,\bar y] [{\bar 4^ + }] [x,0,0] [\bar 4_{{100}}^ +]
45 [\bar z,\bar y,x] [{\bar 4^ + }] [0,y,0] [\bar 4_{{010}}^ +]
46 [\bar z,y,\bar x] m [\bar x,y,x] [{m_{101}}]
47 [z,\bar y,\bar x] [{\bar 4^ - }] [0,y,0] [\bar 4_{{010}}^ -]
48 [z,y,x] m [x,y,x] [{m_{\bar 101}}]