Bilbao Crystallographic Server COREPRESENTATIONS PG

## Irreducible corepresentations of the Magnetic Point Group -31' (N. 17.2.63)

### Table of characters of the unitary symmetry operations

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 GM1+ Ag GM1+ 1 1 1 1 1 1 1 1 1 1 1 1 GM1- Au GM1- 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 GM2+GM3+ 1Eg2Eg GM2+GM3+ 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 GM2-GM3- 1Eu2Eu GM2-GM3- 2 -1 -1 -2 1 1 2 -1 -1 -2 1 1 GM6+GM6+ EgEg GM4GM4 2 -2 -2 2 -2 -2 -2 2 2 -2 2 2 GM4+GM5+ 1Eg2Eg GM5GM6 2 1 1 2 1 1 -2 -1 -1 -2 -1 -1 GM6-GM6- EuEu GM7GM7 2 -2 -2 -2 2 2 -2 2 2 2 -2 -2 GM4-GM5- 1Eu2Eu GM8GM9 2 1 1 -2 -1 -1 -2 -1 -1 2 1 1
 The notation used in this table is an extension to corepresentations of the following notations used for irreducible representations: (1): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press. (2): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302. (3): A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

### Lists of unitary symmetry operations in the conjugacy classes

 C1: 1 C2: 3+001 C3: 3-001 C4: 1 C5: 3+001 C6: 3-001 C7: d1 C8: d3+001 C9: d3-001 C10: d1 C11: d3+001 C12: d3-001

### Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM3+GM2-GM3-GM4GM4GM5GM6GM7GM7GM8GM9
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
2
 ` 0 -1 0 1 -1 0 0 0 1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3+001
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 ` -1 0 0 -1`
 ` e-iπ/3 0 0 eiπ/3`
 ` -1 0 0 -1`
 ` e-iπ/3 0 0 eiπ/3`
3
 ` -1 1 0 -1 0 0 0 0 1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3-001
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 ` -1 0 0 -1`
 ` eiπ/3 0 0 e-iπ/3`
 ` -1 0 0 -1`
 ` eiπ/3 0 0 e-iπ/3`
4
 ` -1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 1`
1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
5
 ` 0 1 0 -1 1 0 0 0 -1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3+001
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 ` -1 0 0 -1`
 ` e-iπ/3 0 0 eiπ/3`
 ` 1 0 0 1`
 ` ei2π/3 0 0 e-i2π/3`
6
 ` 1 -1 0 1 0 0 0 0 -1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3-001
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 ` -1 0 0 -1`
 ` eiπ/3 0 0 e-iπ/3`
 ` 1 0 0 1`
 ` e-i2π/3 0 0 ei2π/3`
7
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
8
 ` 0 -1 0 1 -1 0 0 0 1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 ` 1 0 0 1`
 ` ei2π/3 0 0 e-i2π/3`
 ` 1 0 0 1`
 ` ei2π/3 0 0 e-i2π/3`
9
 ` -1 1 0 -1 0 0 0 0 1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 ` 1 0 0 1`
 ` e-i2π/3 0 0 ei2π/3`
 ` 1 0 0 1`
 ` e-i2π/3 0 0 ei2π/3`
10
 ` -1 0 0 0 -1 0 0 0 -1`
 ` -1 0 0 -1`
d1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` 1 0 0 1`
 ` 1 0 0 1`
11
 ` 0 1 0 -1 1 0 0 0 -1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 ` 1 0 0 1`
 ` ei2π/3 0 0 e-i2π/3`
 ` -1 0 0 -1`
 ` e-iπ/3 0 0 eiπ/3`
12
 ` 1 -1 0 1 0 0 0 0 -1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 ` 1 0 0 1`
 ` e-i2π/3 0 0 ei2π/3`
 ` -1 0 0 -1`
 ` eiπ/3 0 0 e-iπ/3`
13
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1'
 1
 -1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
14
 ` 0 -1 0 1 -1 0 0 0 1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3'+001
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 ` 0 1 -1 0`
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 -1 1 0`
 ` 0 e-iπ/3 e-i2π/3 0`
15
 ` -1 1 0 -1 0 0 0 0 1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3'-001
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 ` 0 1 -1 0`
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 -1 1 0`
 ` 0 eiπ/3 ei2π/3 0`
16
 ` -1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 1`
1'
 1
 1
 ` 0 1 1 0`
 ` 0 1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
17
 ` 0 1 0 -1 1 0 0 0 -1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3'+001
 1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 -1 0`
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 1 -1 0`
 ` 0 ei2π/3 eiπ/3 0`
18
 ` 1 -1 0 1 0 0 0 0 -1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3'-001
 1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 1 -1 0`
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 1 -1 0`
 ` 0 e-i2π/3 e-iπ/3 0`
19
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1'
 1
 -1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
20
 ` 0 -1 0 1 -1 0 0 0 1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3'+001
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 ` 0 -1 1 0`
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 1 -1 0`
 ` 0 ei2π/3 eiπ/3 0`
21
 ` -1 1 0 -1 0 0 0 0 1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3'-001
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 ` 0 -1 1 0`
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 1 -1 0`
 ` 0 e-i2π/3 e-iπ/3 0`
22
 ` -1 0 0 0 -1 0 0 0 -1`
 ` -1 0 0 -1`
d1'
 1
 1
 ` 0 1 1 0`
 ` 0 1 1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
23
 ` 0 1 0 -1 1 0 0 0 -1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3'+001
 1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 1 0`
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 -1 1 0`
 ` 0 e-iπ/3 e-i2π/3 0`
24
 ` 1 -1 0 1 0 0 0 0 -1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3'-001
 1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 -1 1 0`
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 -1 1 0`
 ` 0 eiπ/3 ei2π/3 0`
k-Subgroupsmag
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