Bilbao Crystallographic Server COREPRESENTATIONS PG

## Irreducible corepresentations of the Magnetic Point Group 6'/m'mm' (N. 27.5.104)

### 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 0 2 -1 0 0 2 -1 2 -1 GM3- Eu GM3- 2 -1 0 0 -2 1 0 0 2 -1 -2 1 GM4+ 2Eg GM4 1 -1 -i i 1 -1 -i i -1 1 -1 1 GM5+ 1Eg GM5 1 -1 i -i 1 -1 i -i -1 1 -1 1 GM4- 2Eu GM6 1 -1 -i i -1 1 i -i -1 1 1 -1 GM5- 1Eu GM7 1 -1 i -i -1 1 -i i -1 1 1 -1 GM6+ E1g GM8 2 1 0 0 2 1 0 0 -2 -1 -2 -1 GM6- E1u GM9 2 1 0 0 -2 -1 0 0 -2 -1 2 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, 3-001 C3: 2110, 2010, d2100 C4: 2100, d2110, d2010 C5: 1 C6: 3+001, 3-001 C7: m110, m010, dm100 C8: m100, dm110, dm010 C9: d1 C10: d3+001, d3-001 C11: d1 C12: d3+001, d3-001

### 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 -(1+i√3)/2 (1-i√3)/2 0`
2110
 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 -1 1 0`
2100
 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`
6
 ` -1 0 0 -1 1 0 0 0 -1`
 ` 0 -(1-i√3)/2 (1+i√3)/2 0`
2010
 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 -(1+i√3)/2 (1-i√3)/2 0`
m110
 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 -1 1 0`
m100
 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`
12
 ` 1 0 0 1 -1 0 0 0 1`
 ` 0 -(1-i√3)/2 (1+i√3)/2 0`
m010
 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 (1+i√3)/2 -(1-i√3)/2 0`
d2110
 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 1 -1 0`
d2100
 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`
18
 ` -1 0 0 -1 1 0 0 0 -1`
 ` 0 (1-i√3)/2 -(1+i√3)/2 0`
d2010
 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 (1+i√3)/2 -(1-i√3)/2 0`
dm110
 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 1 -1 0`
dm100
 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`
24
 ` 1 0 0 1 -1 0 0 0 1`
 ` 0 (1-i√3)/2 -(1+i√3)/2 0`
dm010
 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`
25
 ` -1 0 0 0 -1 0 0 0 1`
 ` -i 0 0 i`
2'001
 1
 1
 1
 1
 ` 0 -i -i 0`
 ` 0 -i -i 0`
 1
 1
 1
 1
 ` 0 -i -i 0`
 ` 0 -i -i 0`
26
 ` 0 1 0 -1 1 0 0 0 1`
 ` (√3-i)/2 0 0 (√3+i)/2`
6'-001
 1
 1
 1
 1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 eiπ/6 ei5π/6 0`
 -1
 -1
 -1
 -1
 ` 0 e-i5π/6 e-iπ/6 0`
 ` 0 e-i5π/6 e-iπ/6 0`
27
 ` 1 -1 0 1 0 0 0 0 1`
 ` (√3+i)/2 0 0 (√3-i)/2`
6'+001
 1
 1
 1
 1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 ei5π/6 eiπ/6 0`
 1
 1
 1
 1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 ei5π/6 eiπ/6 0`
28
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
2'1-10
 1
 1
 -1
 -1
 ` -i 0 0 -i`
 ` -i 0 0 -i`
 i
 -i
 i
 -i
 ` -i 0 0 i`
 ` -i 0 0 i`
29
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
2'120
 1
 1
 -1
 -1
 ` ei5π/6 0 0 eiπ/6`
 ` ei5π/6 0 0 eiπ/6`
 i
 -i
 i
 -i
 ` ei5π/6 0 0 e-i5π/6`
 ` ei5π/6 0 0 e-i5π/6`
30
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
2'210
 1
 1
 -1
 -1
 ` eiπ/6 0 0 ei5π/6`
 ` eiπ/6 0 0 ei5π/6`
 i
 -i
 i
 -i
 ` eiπ/6 0 0 e-iπ/6`
 ` eiπ/6 0 0 e-iπ/6`
31
 ` 1 0 0 0 1 0 0 0 -1`
 ` -i 0 0 i`
m'001
 1
 -1
 1
 -1
 ` 0 -i -i 0`
 ` 0 i i 0`
 1
 1
 -1
 -1
 ` 0 -i -i 0`
 ` 0 i i 0`
32
 ` 0 -1 0 1 -1 0 0 0 -1`
 ` (√3-i)/2 0 0 (√3+i)/2`
6'-001
 1
 -1
 1
 -1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 e-i5π/6 e-iπ/6 0`
 -1
 -1
 1
 1
 ` 0 e-i5π/6 e-iπ/6 0`
 ` 0 eiπ/6 ei5π/6 0`
33
 ` -1 1 0 -1 0 0 0 0 -1`
 ` (√3+i)/2 0 0 (√3-i)/2`
6'+001
 1
 -1
 1
 -1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 e-iπ/6 e-i5π/6 0`
 1
 1
 -1
 -1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 e-iπ/6 e-i5π/6 0`
34
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
m'1-10
 1
 -1
 -1
 1
 ` -i 0 0 -i`
 ` i 0 0 i`
 i
 -i
 -i
 i
 ` -i 0 0 i`
 ` i 0 0 -i`
35
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 -i -i 0`
m'120
 1
 -1
 -1
 1
 ` ei5π/6 0 0 eiπ/6`
 ` e-iπ/6 0 0 e-i5π/6`
 i
 -i
 -i
 i
 ` ei5π/6 0 0 e-i5π/6`
 ` e-iπ/6 0 0 eiπ/6`
36
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
m'210
 1
 -1
 -1
 1
 ` eiπ/6 0 0 ei5π/6`
 ` e-i5π/6 0 0 e-iπ/6`
 i
 -i
 -i
 i
 ` eiπ/6 0 0 e-iπ/6`
 ` e-i5π/6 0 0 ei5π/6`
37
 ` -1 0 0 0 -1 0 0 0 1`
 ` i 0 0 -i`
d2'001
 1
 1
 1
 1
 ` 0 -i -i 0`
 ` 0 -i -i 0`
 -1
 -1
 -1
 -1
 ` 0 i i 0`
 ` 0 i i 0`
38
 ` 0 1 0 -1 1 0 0 0 1`
 ` -(√3-i)/2 0 0 -(√3+i)/2`
d6'-001
 1
 1
 1
 1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 eiπ/6 ei5π/6 0`
 1
 1
 1
 1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 eiπ/6 ei5π/6 0`
39
 ` 1 -1 0 1 0 0 0 0 1`
 ` -(√3+i)/2 0 0 -(√3-i)/2`
d6'+001
 1
 1
 1
 1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 ei5π/6 eiπ/6 0`
 -1
 -1
 -1
 -1
 ` 0 e-iπ/6 e-i5π/6 0`
 ` 0 e-iπ/6 e-i5π/6 0`
40
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
d2'1-10
 1
 1
 -1
 -1
 ` -i 0 0 -i`
 ` -i 0 0 -i`
 -i
 i
 -i
 i
 ` i 0 0 -i`
 ` i 0 0 -i`
41
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 i i 0`
d2'120
 1
 1
 -1
 -1
 ` ei5π/6 0 0 eiπ/6`
 ` ei5π/6 0 0 eiπ/6`
 -i
 i
 -i
 i
 ` e-iπ/6 0 0 eiπ/6`
 ` e-iπ/6 0 0 eiπ/6`
42
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
d2'210
 1
 1
 -1
 -1
 ` eiπ/6 0 0 ei5π/6`
 ` eiπ/6 0 0 ei5π/6`
 -i
 i
 -i
 i
 ` e-i5π/6 0 0 ei5π/6`
 ` e-i5π/6 0 0 ei5π/6`
43
 ` 1 0 0 0 1 0 0 0 -1`
 ` i 0 0 -i`
dm'001
 1
 -1
 1
 -1
 ` 0 -i -i 0`
 ` 0 i i 0`
 -1
 -1
 1
 1
 ` 0 i i 0`
 ` 0 -i -i 0`
44
 ` 0 -1 0 1 -1 0 0 0 -1`
 ` -(√3-i)/2 0 0 -(√3+i)/2`
d6'-001
 1
 -1
 1
 -1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 e-i5π/6 e-iπ/6 0`
 1
 1
 -1
 -1
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 e-i5π/6 e-iπ/6 0`
45
 ` -1 1 0 -1 0 0 0 0 -1`
 ` -(√3+i)/2 0 0 -(√3-i)/2`
d6'+001
 1
 -1
 1
 -1
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 e-iπ/6 e-i5π/6 0`
 -1
 -1
 1
 1
 ` 0 e-iπ/6 e-i5π/6 0`
 ` 0 ei5π/6 eiπ/6 0`
46
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
dm'1-10
 1
 -1
 -1
 1
 ` -i 0 0 -i`
 ` i 0 0 i`
 -i
 i
 i
 -i
 ` i 0 0 -i`
 ` -i 0 0 i`
47
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 i i 0`
dm'120
 1
 -1
 -1
 1
 ` ei5π/6 0 0 eiπ/6`
 ` e-iπ/6 0 0 e-i5π/6`
 -i
 i
 i
 -i
 ` e-iπ/6 0 0 eiπ/6`
 ` ei5π/6 0 0 e-i5π/6`
48
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
dm'210
 1
 -1
 -1
 1
 ` eiπ/6 0 0 ei5π/6`
 ` e-i5π/6 0 0 e-iπ/6`
 -i
 i
 i
 -i
 ` e-i5π/6 0 0 ei5π/6`
 ` eiπ/6 0 0 e-iπ/6`
k-Subgroupsmag
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