Bilbao Crystallographic Server COREPRESENTATIONS PG

## Irreducible corepresentations of the Magnetic Point Group -3m (N. 20.1.71)

### Table of characters of the unitary symmetry operations

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 GM1+ A1g GM1+ 1 1 1 1 1 1 1 1 1 1 1 1 GM1- A1u GM1- 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 GM2+ A2g GM2+ 1 1 -1 1 1 -1 1 1 -1 1 1 -1 GM2- A2u GM2- 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 GM3+ Eg GM3+ 2 -1 0 2 -1 0 2 -1 0 2 -1 0 GM3- Eu GM3- 2 -1 0 -2 1 0 2 -1 0 -2 1 0 GM4+ 2Eg GM4 1 -1 -i 1 -1 -i -1 1 i -1 1 i GM5+ 1Eg GM5 1 -1 i 1 -1 i -1 1 -i -1 1 -i GM4- 2Eu GM6 1 -1 -i -1 1 i -1 1 i 1 -1 -i GM5- 1Eu GM7 1 -1 i -1 1 -i -1 1 -i 1 -1 i GM6+ E1g GM8 2 1 0 2 1 0 -2 -1 0 -2 -1 0 GM6- E1u GM9 2 1 0 -2 -1 0 -2 -1 0 2 1 0
 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, 3-001 C3: 2110, 2120, 2210 C4: 1 C5: 3+001, 3-001 C6: m110, m120, m210 C7: d1 C8: d3+001, d3-001 C9: d2110, d2120, d2210 C10: d1 C11: d3+001, d3-001 C12: dm110, dm120, dm210

### Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM3-GM4GM5GM6GM7GM8GM9
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 1
 1
 1
 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
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 -1
 -1
 -1
 -1
 ` e-iπ/3 0 0 eiπ/3`
 ` 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
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 -1
 -1
 -1
 -1
 ` eiπ/3 0 0 e-iπ/3`
 ` eiπ/3 0 0 e-iπ/3`
4
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
21-10
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 -i
 i
 -i
 i
 ` 0 -1 1 0`
 ` 0 -1 1 0`
5
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
2120
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 -i
 i
 -i
 i
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 eiπ/3 ei2π/3 0`
6
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
2210
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 -i
 i
 -i
 i
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 e-iπ/3 e-i2π/3 0`
7
 ` -1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 1`
1
 1
 -1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 1
 1
 -1
 -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`
3+001
 1
 -1
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 -1
 -1
 1
 1
 ` e-iπ/3 0 0 eiπ/3`
 ` 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`
3-001
 1
 -1
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 -1
 -1
 1
 1
 ` eiπ/3 0 0 e-iπ/3`
 ` e-i2π/3 0 0 ei2π/3`
10
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
m1-10
 1
 -1
 -1
 1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 -i
 i
 i
 -i
 ` 0 -1 1 0`
 ` 0 1 -1 0`
11
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 -i -i 0`
m120
 1
 -1
 -1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 -i
 i
 i
 -i
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 e-i2π/3 e-iπ/3 0`
12
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
m210
 1
 -1
 -1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 -i
 i
 i
 -i
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 ei2π/3 eiπ/3 0`
13
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 -1
 -1
 -1
 -1
 ` -1 0 0 -1`
 ` -1 0 0 -1`
14
 ` 0 -1 0 1 -1 0 0 0 1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
15
 ` -1 1 0 -1 0 0 0 0 1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
16
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
d21-10
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 i
 -i
 i
 -i
 ` 0 1 -1 0`
 ` 0 1 -1 0`
17
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 i i 0`
d2120
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 i
 -i
 i
 -i
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 e-i2π/3 e-iπ/3 0`
18
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
d2210
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 i
 -i
 i
 -i
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 ei2π/3 eiπ/3 0`
19
 ` -1 0 0 0 -1 0 0 0 -1`
 ` -1 0 0 -1`
d1
 1
 -1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 -1
 -1
 1
 1
 ` -1 0 0 -1`
 ` 1 0 0 1`
20
 ` 0 1 0 -1 1 0 0 0 -1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 -1
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 1
 1
 -1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
21
 ` 1 -1 0 1 0 0 0 0 -1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 -1
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 1
 1
 -1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
22
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
dm1-10
 1
 -1
 -1
 1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 i
 -i
 -i
 i
 ` 0 1 -1 0`
 ` 0 -1 1 0`
23
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 i i 0`
dm120
 1
 -1
 -1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 i
 -i
 -i
 i
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 eiπ/3 ei2π/3 0`
24
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
dm210
 1
 -1
 -1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 i
 -i
 -i
 i
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 e-iπ/3 e-i2π/3 0`
k-Subgroupsmag
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