Bilbao Crystallographic Server COREPRESENTATIONS PG

## Irreducible corepresentations of the Magnetic Point Group 23 (N. 28.1.107)

### Table of characters of the unitary symmetry operations

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 GM1 A GM1 1 1 1 1 1 1 1 GM2 1E GM2 1 1 -(1-i√3)/2 -(1+i√3)/2 1 -(1-i√3)/2 -(1+i√3)/2 GM3 2E GM3 1 1 -(1+i√3)/2 -(1-i√3)/2 1 -(1+i√3)/2 -(1-i√3)/2 GM4 T GM4 3 -1 0 0 3 0 0 GM5 E GM5 2 0 1 1 -2 -1 -1 GM7 2F GM6 2 0 -(1-i√3)/2 -(1+i√3)/2 -2 (1-i√3)/2 (1+i√3)/2 GM6 1F GM7 2 0 -(1+i√3)/2 -(1-i√3)/2 -2 (1+i√3)/2 (1-i√3)/2
 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: 2001, 2010, 2100, d2001, d2010, d2100 C3: 3+111, 3+111, 3+111, 3+111 C4: 3-111, 3-111, 3-111, 3-111 C5: d1 C6: d3+111, d3+111, d3+111, d3+111 C7: d3-111, d3-111, d3-111, d3-111

### Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
2
 ` -1 0 0 0 -1 0 0 0 1`
 ` -i 0 0 i`
2001
 1
 1
 1
 ` 1 0 0 0 -1 0 0 0 -1`
 ` -i 0 0 i`
 ` -i 0 0 i`
 ` -i 0 0 i`
3
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 -1 1 0`
2010
 1
 1
 1
 ` -1 0 0 0 -1 0 0 0 1`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
4
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 -i -i 0`
2100
 1
 1
 1
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
 ` 0 -i -i 0`
 ` 0 -i -i 0`
5
 ` 0 0 1 1 0 0 0 1 0`
 ` (1-i)/2 -(1+i)/2 (1-i)/2 (1+i)/2`
3+111
 1
 ei2π/3
 e-i2π/3
 ` 0 0 1 1 0 0 0 1 0`
 ` e-iπ/4√2/2 e-i3π/4√2/2 e-iπ/4√2/2 eiπ/4√2/2`
 ` ei5π/12√2/2 e-iπ/12√2/2 ei5π/12√2/2 ei11π/12√2/2`
 ` e-i11π/12√2/2 ei7π/12√2/2 e-i11π/12√2/2 e-i5π/12√2/2`
6
 ` 0 0 1 -1 0 0 0 -1 0`
 ` (1+i)/2 -(1-i)/2 (1+i)/2 (1-i)/2`
3+-11-1
 1
 ei2π/3
 e-i2π/3
 ` 0 0 -1 1 0 0 0 -1 0`
 ` eiπ/4√2/2 ei3π/4√2/2 eiπ/4√2/2 e-iπ/4√2/2`
 ` ei11π/12√2/2 e-i7π/12√2/2 ei11π/12√2/2 ei5π/12√2/2`
 ` e-i5π/12√2/2 eiπ/12√2/2 e-i5π/12√2/2 e-i11π/12√2/2`
7
 ` 0 0 -1 -1 0 0 0 1 0`
 ` (1+i)/2 (1-i)/2 -(1+i)/2 (1-i)/2`
3+1-1-1
 1
 ei2π/3
 e-i2π/3
 ` 0 0 1 -1 0 0 0 -1 0`
 ` eiπ/4√2/2 e-iπ/4√2/2 e-i3π/4√2/2 e-iπ/4√2/2`
 ` ei11π/12√2/2 ei5π/12√2/2 e-iπ/12√2/2 ei5π/12√2/2`
 ` e-i5π/12√2/2 e-i11π/12√2/2 ei7π/12√2/2 e-i11π/12√2/2`
8
 ` 0 0 -1 1 0 0 0 -1 0`
 ` (1-i)/2 (1+i)/2 -(1-i)/2 (1+i)/2`
3+-1-11
 1
 ei2π/3
 e-i2π/3
 ` 0 0 -1 -1 0 0 0 1 0`
 ` e-iπ/4√2/2 eiπ/4√2/2 ei3π/4√2/2 eiπ/4√2/2`
 ` ei5π/12√2/2 ei11π/12√2/2 e-i7π/12√2/2 ei11π/12√2/2`
 ` e-i11π/12√2/2 e-i5π/12√2/2 eiπ/12√2/2 e-i5π/12√2/2`
9
 ` 0 1 0 0 0 1 1 0 0`
 ` (1+i)/2 (1+i)/2 -(1-i)/2 (1-i)/2`
3-111
 1
 e-i2π/3
 ei2π/3
 ` 0 1 0 0 0 1 1 0 0`
 ` eiπ/4√2/2 eiπ/4√2/2 ei3π/4√2/2 e-iπ/4√2/2`
 ` e-i5π/12√2/2 e-i5π/12√2/2 eiπ/12√2/2 e-i11π/12√2/2`
 ` ei11π/12√2/2 ei11π/12√2/2 e-i7π/12√2/2 ei5π/12√2/2`
10
 ` 0 -1 0 0 0 1 -1 0 0`
 ` (1-i)/2 -(1-i)/2 (1+i)/2 (1+i)/2`
3-1-1-1
 1
 e-i2π/3
 ei2π/3
 ` 0 -1 0 0 0 -1 1 0 0`
 ` e-iπ/4√2/2 ei3π/4√2/2 eiπ/4√2/2 eiπ/4√2/2`
 ` e-i11π/12√2/2 eiπ/12√2/2 e-i5π/12√2/2 e-i5π/12√2/2`
 ` ei5π/12√2/2 e-i7π/12√2/2 ei11π/12√2/2 ei11π/12√2/2`
11
 ` 0 1 0 0 0 -1 -1 0 0`
 ` (1+i)/2 -(1+i)/2 (1-i)/2 (1-i)/2`
3--1-11
 1
 e-i2π/3
 ei2π/3
 ` 0 -1 0 0 0 1 -1 0 0`
 ` eiπ/4√2/2 e-i3π/4√2/2 e-iπ/4√2/2 e-iπ/4√2/2`
 ` e-i5π/12√2/2 ei7π/12√2/2 e-i11π/12√2/2 e-i11π/12√2/2`
 ` ei11π/12√2/2 e-iπ/12√2/2 ei5π/12√2/2 ei5π/12√2/2`
12
 ` 0 -1 0 0 0 -1 1 0 0`
 ` (1-i)/2 (1-i)/2 -(1+i)/2 (1+i)/2`
3--11-1
 1
 e-i2π/3
 ei2π/3
 ` 0 1 0 0 0 -1 -1 0 0`
 ` e-iπ/4√2/2 e-iπ/4√2/2 e-i3π/4√2/2 eiπ/4√2/2`
 ` e-i11π/12√2/2 e-i11π/12√2/2 ei7π/12√2/2 e-i5π/12√2/2`
 ` ei5π/12√2/2 ei5π/12√2/2 e-iπ/12√2/2 ei11π/12√2/2`
13
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 1
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
14
 ` -1 0 0 0 -1 0 0 0 1`
 ` i 0 0 -i`
d2001
 1
 1
 1
 ` 1 0 0 0 -1 0 0 0 -1`
 ` i 0 0 -i`
 ` i 0 0 -i`
 ` i 0 0 -i`
15
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 1 -1 0`
d2010
 1
 1
 1
 ` -1 0 0 0 -1 0 0 0 1`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
16
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 i i 0`
d2100
 1
 1
 1
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 i i 0`
 ` 0 i i 0`
 ` 0 i i 0`
17
 ` 0 0 1 1 0 0 0 1 0`
 ` -(1-i)/2 (1+i)/2 -(1-i)/2 -(1+i)/2`
d3+111
 1
 ei2π/3
 e-i2π/3
 ` 0 0 1 1 0 0 0 1 0`
 ` ei3π/4√2/2 eiπ/4√2/2 ei3π/4√2/2 e-i3π/4√2/2`
 ` e-i7π/12√2/2 ei11π/12√2/2 e-i7π/12√2/2 e-iπ/12√2/2`
 ` eiπ/12√2/2 e-i5π/12√2/2 eiπ/12√2/2 ei7π/12√2/2`
18
 ` 0 0 1 -1 0 0 0 -1 0`
 ` -(1+i)/2 (1-i)/2 -(1+i)/2 -(1-i)/2`
d3+-11-1
 1
 ei2π/3
 e-i2π/3
 ` 0 0 -1 1 0 0 0 -1 0`
 ` e-i3π/4√2/2 e-iπ/4√2/2 e-i3π/4√2/2 ei3π/4√2/2`
 ` e-iπ/12√2/2 ei5π/12√2/2 e-iπ/12√2/2 e-i7π/12√2/2`
 ` ei7π/12√2/2 e-i11π/12√2/2 ei7π/12√2/2 eiπ/12√2/2`
19
 ` 0 0 -1 -1 0 0 0 1 0`
 ` -(1+i)/2 -(1-i)/2 (1+i)/2 -(1-i)/2`
d3+1-1-1
 1
 ei2π/3
 e-i2π/3
 ` 0 0 1 -1 0 0 0 -1 0`
 ` e-i3π/4√2/2 ei3π/4√2/2 eiπ/4√2/2 ei3π/4√2/2`
 ` e-iπ/12√2/2 e-i7π/12√2/2 ei11π/12√2/2 e-i7π/12√2/2`
 ` ei7π/12√2/2 eiπ/12√2/2 e-i5π/12√2/2 eiπ/12√2/2`
20
 ` 0 0 -1 1 0 0 0 -1 0`
 ` -(1-i)/2 -(1+i)/2 (1-i)/2 -(1+i)/2`
d3+-1-11
 1
 ei2π/3
 e-i2π/3
 ` 0 0 -1 -1 0 0 0 1 0`
 ` ei3π/4√2/2 e-i3π/4√2/2 e-iπ/4√2/2 e-i3π/4√2/2`
 ` e-i7π/12√2/2 e-iπ/12√2/2 ei5π/12√2/2 e-iπ/12√2/2`
 ` eiπ/12√2/2 ei7π/12√2/2 e-i11π/12√2/2 ei7π/12√2/2`
21
 ` 0 1 0 0 0 1 1 0 0`
 ` -(1+i)/2 -(1+i)/2 (1-i)/2 -(1-i)/2`
d3-111
 1
 e-i2π/3
 ei2π/3
 ` 0 1 0 0 0 1 1 0 0`
 ` e-i3π/4√2/2 e-i3π/4√2/2 e-iπ/4√2/2 ei3π/4√2/2`
 ` ei7π/12√2/2 ei7π/12√2/2 e-i11π/12√2/2 eiπ/12√2/2`
 ` e-iπ/12√2/2 e-iπ/12√2/2 ei5π/12√2/2 e-i7π/12√2/2`
22
 ` 0 -1 0 0 0 1 -1 0 0`
 ` -(1-i)/2 (1-i)/2 -(1+i)/2 -(1+i)/2`
d3-1-1-1
 1
 e-i2π/3
 ei2π/3
 ` 0 -1 0 0 0 -1 1 0 0`
 ` ei3π/4√2/2 e-iπ/4√2/2 e-i3π/4√2/2 e-i3π/4√2/2`
 ` eiπ/12√2/2 e-i11π/12√2/2 ei7π/12√2/2 ei7π/12√2/2`
 ` e-i7π/12√2/2 ei5π/12√2/2 e-iπ/12√2/2 e-iπ/12√2/2`
23
 ` 0 1 0 0 0 -1 -1 0 0`
 ` -(1+i)/2 (1+i)/2 -(1-i)/2 -(1-i)/2`
d3--1-11
 1
 e-i2π/3
 ei2π/3
 ` 0 -1 0 0 0 1 -1 0 0`
 ` e-i3π/4√2/2 eiπ/4√2/2 ei3π/4√2/2 ei3π/4√2/2`
 ` ei7π/12√2/2 e-i5π/12√2/2 eiπ/12√2/2 eiπ/12√2/2`
 ` e-iπ/12√2/2 ei11π/12√2/2 e-i7π/12√2/2 e-i7π/12√2/2`
24
 ` 0 -1 0 0 0 -1 1 0 0`
 ` -(1-i)/2 -(1-i)/2 (1+i)/2 -(1+i)/2`
d3--11-1
 1
 e-i2π/3
 ei2π/3
 ` 0 1 0 0 0 -1 -1 0 0`
 ` ei3π/4√2/2 ei3π/4√2/2 eiπ/4√2/2 e-i3π/4√2/2`
 ` eiπ/12√2/2 eiπ/12√2/2 e-i5π/12√2/2 ei7π/12√2/2`
 ` e-i7π/12√2/2 e-i7π/12√2/2 ei11π/12√2/2 e-iπ/12√2/2`
k-Subgroupsmag
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