Bilbao Crystallographic Server COREPRESENTATIONS PG

## Irreducible corepresentations of the Magnetic Point Group 432 (N. 30.1.112)

### Table of characters of the unitary symmetry operations

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 GM1 A1 GM1 1 1 1 1 1 1 1 1 GM2 A2 GM2 1 1 1 -1 -1 1 1 -1 GM3 E GM3 2 2 -1 0 0 2 -1 0 GM4 T1 GM4 3 -1 0 -1 1 3 0 1 GM5 T2 GM5 3 -1 0 1 -1 3 0 -1 GM6 E1 GM6 2 0 1 0 √2 -2 -1 -√2 GM7 E2 GM7 2 0 1 0 -√2 -2 -1 √2 GM8 F GM8 4 0 -1 0 0 -4 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: 2001, 2010, 2100, d2001, d2010, d2100 C3: 3+111, 3+111, 3+111, 3+111, 3-111, 3-111, 3-111, 3-111 C4: 2110, 2110, 2011, 2011, 2101, 2101, d2110, d2110, d2011, d2011, d2101, d2101 C5: 4-001, 4+001, 4-100, 4+100, 4+010, 4-010 C6: d1 C7: d3+111, d3+111, d3+111, d3+111, d3-111, d3-111, d3-111, d3-111 C8: d4-001, d4+001, d4-100, d4+100, d4+010, d4-010

### Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7GM8
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 1`
 ` 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1`
2
 ` -1 0 0 0 -1 0 0 0 1`
 ` -i 0 0 i`
2001
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 0 -1 0 0 0 -1`
 ` -i 0 0 i`
 ` -i 0 0 i`
 ` -i 0 0 0 0 i 0 0 0 0 i 0 0 0 0 -i`
3
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 -1 1 0`
2010
 1
 1
 ` 1 0 0 1`
 ` -1 0 0 0 -1 0 0 0 1`
 ` -1 0 0 0 -1 0 0 0 1`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 0 0 1 0 0 0 0 0 0 -i 0 0 -i 0`
4
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 -i -i 0`
2100
 1
 1
 ` 1 0 0 1`
 ` -1 0 0 0 1 0 0 0 -1`
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
 ` 0 -i -i 0`
 ` 0 -i 0 0 -i 0 0 0 0 0 0 -1 0 0 1 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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 1 1 0 0 0 1 0`
 ` 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`
 ` 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 0 0 ei5π/12√2/2 ei11π/12√2/2 0 0 0 0 e-i5π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 e-i11π/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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 -1 1 0 0 0 -1 0`
 ` 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`
 ` 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 0 0 ei11π/12√2/2 ei5π/12√2/2 0 0 0 0 e-i11π/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 e-i5π/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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 1 -1 0 0 0 -1 0`
 ` 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`
 ` 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 0 0 e-iπ/12√2/2 ei5π/12√2/2 0 0 0 0 e-i11π/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 e-i5π/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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 -1 -1 0 0 0 1 0`
 ` 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`
 ` 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 0 0 e-i7π/12√2/2 ei11π/12√2/2 0 0 0 0 e-i5π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 e-i11π/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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 1 0 0 0 1 1 0 0`
 ` 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`
 ` 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 0 0 eiπ/12√2/2 e-i11π/12√2/2 0 0 0 0 ei5π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 ei11π/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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 -1 0 0 0 -1 1 0 0`
 ` 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-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 0 0 e-i5π/12√2/2 e-i5π/12√2/2 0 0 0 0 ei11π/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 ei5π/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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 -1 0 0 0 1 -1 0 0`
 ` 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`
 ` 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 0 0 e-i11π/12√2/2 e-i11π/12√2/2 0 0 0 0 ei5π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 ei11π/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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 1 0 0 0 -1 -1 0 0`
 ` 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-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 0 0 ei7π/12√2/2 e-i5π/12√2/2 0 0 0 0 ei11π/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 ei5π/12√2/2`
13
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 -(1+i)√2/2 (1-i)√2/2 0`
2110
 1
 -1
 ` 0 1 1 0`
 ` -1 0 0 0 0 1 0 1 0`
 ` 1 0 0 0 0 -1 0 -1 0`
 ` 0 e-i3π/4 e-iπ/4 0`
 ` 0 eiπ/4 ei3π/4 0`
 ` 0 0 -1 0 0 0 0 -1 1 0 0 0 0 1 0 0`
14
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 -(1-i)√2/2 (1+i)√2/2 0`
21-10
 1
 -1
 ` 0 1 1 0`
 ` -1 0 0 0 0 -1 0 -1 0`
 ` 1 0 0 0 0 1 0 1 0`
 ` 0 ei3π/4 eiπ/4 0`
 ` 0 e-iπ/4 e-i3π/4 0`
 ` 0 0 i 0 0 0 0 -i i 0 0 0 0 -i 0 0`
15
 ` 0 1 0 -1 0 0 0 0 1`
 ` (1+i)√2/2 0 0 (1-i)√2/2`
4-001
 1
 -1
 ` 0 1 1 0`
 ` 1 0 0 0 0 1 0 -1 0`
 ` -1 0 0 0 0 -1 0 1 0`
 ` eiπ/4 0 0 e-iπ/4`
 ` e-i3π/4 0 0 ei3π/4`
 ` 0 0 0 -i 0 0 -i 0 0 1 0 0 -1 0 0 0`
16
 ` 0 -1 0 1 0 0 0 0 1`
 ` (1-i)√2/2 0 0 (1+i)√2/2`
4+001
 1
 -1
 ` 0 1 1 0`
 ` 1 0 0 0 0 -1 0 1 0`
 ` -1 0 0 0 0 1 0 -1 0`
 ` e-iπ/4 0 0 eiπ/4`
 ` ei3π/4 0 0 e-i3π/4`
 ` 0 0 0 -1 0 0 1 0 0 i 0 0 i 0 0 0`
17
 ` 1 0 0 0 0 1 0 -1 0`
 ` √2/2 i√2/2 i√2/2 √2/2`
4-100
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 -1 0 1 0 1 0 0`
 ` 0 0 1 0 -1 0 -1 0 0`
 ` 1/√2 i/√2 i/√2 1/√2`
 ` -1/√2 -i/√2 -i/√2 -1/√2`
 ` 0 0 e-i5π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 e-i11π/12√2/2 e-i7π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 e-iπ/12√2/2 0 0`
18
 ` -1 0 0 0 0 1 0 1 0`
 ` i√2/2 √2/2 -√2/2 -i√2/2`
2011
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 1 0 -1 0 1 0 0`
 ` 0 0 -1 0 1 0 -1 0 0`
 ` i/√2 1/√2 -1/√2 -i/√2`
 ` -i/√2 -1/√2 1/√2 i/√2`
 ` 0 0 e-i11π/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 e-i5π/12√2/2 e-iπ/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 e-i7π/12√2/2 0 0`
19
 ` -1 0 0 0 0 -1 0 -1 0`
 ` -i√2/2 √2/2 -√2/2 i√2/2`
201-1
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 -1 0 -1 0 -1 0 0`
 ` 0 0 1 0 1 0 1 0 0`
 ` -i/√2 1/√2 -1/√2 i/√2`
 ` i/√2 -1/√2 1/√2 -i/√2`
 ` 0 0 e-i11π/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 e-i5π/12√2/2 e-iπ/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 e-i7π/12√2/2 0 0`
20
 ` 1 0 0 0 0 -1 0 1 0`
 ` √2/2 -i√2/2 -i√2/2 √2/2`
4+100
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 1 0 1 0 -1 0 0`
 ` 0 0 -1 0 -1 0 1 0 0`
 ` 1/√2 -i/√2 -i/√2 1/√2`
 ` -1/√2 i/√2 i/√2 -1/√2`
 ` 0 0 ei7π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 eiπ/12√2/2 ei5π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 ei11π/12√2/2 0 0`
21
 ` 0 0 1 0 1 0 -1 0 0`
 ` √2/2 -√2/2 √2/2 √2/2`
4+010
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 0 1 0 0 0 0 1`
 ` 0 1 0 -1 0 0 0 0 -1`
 ` 1/√2 -1/√2 1/√2 1/√2`
 ` -1/√2 1/√2 -1/√2 -1/√2`
 ` 0 0 e-i7π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 e-iπ/12√2/2 e-i5π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 e-i11π/12√2/2 0 0`
22
 ` 0 0 1 0 -1 0 1 0 0`
 ` -i√2/2 -i√2/2 -i√2/2 i√2/2`
2101
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 -1 0 -1 0 0 0 0 1`
 ` -i/√2 -i/√2 -i/√2 i/√2`
 ` i/√2 i/√2 i/√2 -i/√2`
 ` 0 0 e-iπ/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 e-i7π/12√2/2 e-i11π/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 e-i5π/12√2/2 0 0`
23
 ` 0 0 -1 0 1 0 1 0 0`
 ` √2/2 √2/2 -√2/2 √2/2`
4-010
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 0 -1 0 0 0 0 1`
 ` 0 -1 0 1 0 0 0 0 -1`
 ` 1/√2 1/√2 -1/√2 1/√2`
 ` -1/√2 -1/√2 1/√2 -1/√2`
 ` 0 0 ei5π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 ei11π/12√2/2 ei7π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 eiπ/12√2/2 0 0`
24
 ` 0 0 -1 0 -1 0 -1 0 0`
 ` i√2/2 -i√2/2 -i√2/2 -i√2/2`
2-101
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 1 0 1 0 0 0 0 1`
 ` i/√2 -i/√2 -i/√2 -i/√2`
 ` -i/√2 i/√2 i/√2 i/√2`
 ` 0 0 e-iπ/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 e-i7π/12√2/2 e-i11π/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 e-i5π/12√2/2 0 0`
25
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
 ` -1 0 0 -1`
 ` -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1`
26
 ` -1 0 0 0 -1 0 0 0 1`
 ` i 0 0 -i`
d2001
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 0 -1 0 0 0 -1`
 ` i 0 0 -i`
 ` i 0 0 -i`
 ` i 0 0 0 0 -i 0 0 0 0 -i 0 0 0 0 i`
27
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 1 -1 0`
d2010
 1
 1
 ` 1 0 0 1`
 ` -1 0 0 0 -1 0 0 0 1`
 ` -1 0 0 0 -1 0 0 0 1`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 0 0 -1 0 0 0 0 0 0 i 0 0 i 0`
28
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 i i 0`
d2100
 1
 1
 ` 1 0 0 1`
 ` -1 0 0 0 1 0 0 0 -1`
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 i i 0`
 ` 0 i i 0`
 ` 0 i 0 0 i 0 0 0 0 0 0 1 0 0 -1 0`
29
 ` 0 0 1 1 0 0 0 1 0`
 ` -(1-i)/2 (1+i)/2 -(1-i)/2 -(1+i)/2`
d3+111
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 1 1 0 0 0 1 0`
 ` 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`
 ` 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 0 0 e-i7π/12√2/2 e-iπ/12√2/2 0 0 0 0 ei7π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 eiπ/12√2/2`
30
 ` 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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 -1 1 0 0 0 -1 0`
 ` 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-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 0 0 e-iπ/12√2/2 e-i7π/12√2/2 0 0 0 0 eiπ/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 ei7π/12√2/2`
31
 ` 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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 1 -1 0 0 0 -1 0`
 ` 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-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 0 0 ei11π/12√2/2 e-i7π/12√2/2 0 0 0 0 eiπ/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 ei7π/12√2/2`
32
 ` 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
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` 0 0 -1 -1 0 0 0 1 0`
 ` 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`
 ` 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 0 0 ei5π/12√2/2 e-iπ/12√2/2 0 0 0 0 ei7π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 eiπ/12√2/2`
33
 ` 0 1 0 0 0 1 1 0 0`
 ` -(1+i)/2 -(1+i)/2 (1-i)/2 -(1-i)/2`
d3-111
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 1 0 0 0 1 1 0 0`
 ` 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`
 ` 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 0 0 e-i11π/12√2/2 eiπ/12√2/2 0 0 0 0 e-i7π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 e-iπ/12√2/2`
34
 ` 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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 -1 0 0 0 -1 1 0 0`
 ` 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`
 ` 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 0 0 ei7π/12√2/2 ei7π/12√2/2 0 0 0 0 e-iπ/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 e-i7π/12√2/2`
35
 ` 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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 -1 0 0 0 1 -1 0 0`
 ` 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`
 ` 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 0 0 eiπ/12√2/2 eiπ/12√2/2 0 0 0 0 e-i7π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 e-iπ/12√2/2`
36
 ` 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
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` 0 1 0 0 0 -1 -1 0 0`
 ` 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`
 ` 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 0 0 e-i5π/12√2/2 ei7π/12√2/2 0 0 0 0 e-iπ/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 e-i7π/12√2/2`
37
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 (1+i)√2/2 -(1-i)√2/2 0`
d2110
 1
 -1
 ` 0 1 1 0`
 ` -1 0 0 0 0 1 0 1 0`
 ` 1 0 0 0 0 -1 0 -1 0`
 ` 0 eiπ/4 ei3π/4 0`
 ` 0 e-i3π/4 e-iπ/4 0`
 ` 0 0 1 0 0 0 0 1 -1 0 0 0 0 -1 0 0`
38
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 (1-i)√2/2 -(1+i)√2/2 0`
d21-10
 1
 -1
 ` 0 1 1 0`
 ` -1 0 0 0 0 -1 0 -1 0`
 ` 1 0 0 0 0 1 0 1 0`
 ` 0 e-iπ/4 e-i3π/4 0`
 ` 0 ei3π/4 eiπ/4 0`
 ` 0 0 -i 0 0 0 0 i -i 0 0 0 0 i 0 0`
39
 ` 0 1 0 -1 0 0 0 0 1`
 ` -(1+i)√2/2 0 0 -(1-i)√2/2`
d4-001
 1
 -1
 ` 0 1 1 0`
 ` 1 0 0 0 0 1 0 -1 0`
 ` -1 0 0 0 0 -1 0 1 0`
 ` e-i3π/4 0 0 ei3π/4`
 ` eiπ/4 0 0 e-iπ/4`
 ` 0 0 0 i 0 0 i 0 0 -1 0 0 1 0 0 0`
40
 ` 0 -1 0 1 0 0 0 0 1`
 ` -(1-i)√2/2 0 0 -(1+i)√2/2`
d4+001
 1
 -1
 ` 0 1 1 0`
 ` 1 0 0 0 0 -1 0 1 0`
 ` -1 0 0 0 0 1 0 -1 0`
 ` ei3π/4 0 0 e-i3π/4`
 ` e-iπ/4 0 0 eiπ/4`
 ` 0 0 0 1 0 0 -1 0 0 -i 0 0 -i 0 0 0`
41
 ` 1 0 0 0 0 1 0 -1 0`
 ` -√2/2 -i√2/2 -i√2/2 -√2/2`
d4-100
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 -1 0 1 0 1 0 0`
 ` 0 0 1 0 -1 0 -1 0 0`
 ` -1/√2 -i/√2 -i/√2 -1/√2`
 ` 1/√2 i/√2 i/√2 1/√2`
 ` 0 0 ei7π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 eiπ/12√2/2 ei5π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 ei11π/12√2/2 0 0`
42
 ` -1 0 0 0 0 1 0 1 0`
 ` -i√2/2 -√2/2 √2/2 i√2/2`
d2011
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 1 0 -1 0 1 0 0`
 ` 0 0 -1 0 1 0 -1 0 0`
 ` -i/√2 -1/√2 1/√2 i/√2`
 ` i/√2 1/√2 -1/√2 -i/√2`
 ` 0 0 eiπ/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 ei7π/12√2/2 ei11π/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 ei5π/12√2/2 0 0`
43
 ` -1 0 0 0 0 -1 0 -1 0`
 ` i√2/2 -√2/2 √2/2 -i√2/2`
d201-1
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 -1 0 -1 0 -1 0 0`
 ` 0 0 1 0 1 0 1 0 0`
 ` i/√2 -1/√2 1/√2 -i/√2`
 ` -i/√2 1/√2 -1/√2 i/√2`
 ` 0 0 eiπ/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 ei7π/12√2/2 ei11π/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 ei5π/12√2/2 0 0`
44
 ` 1 0 0 0 0 -1 0 1 0`
 ` -√2/2 i√2/2 i√2/2 -√2/2`
d4+100
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 0 1 0 1 0 -1 0 0`
 ` 0 0 -1 0 -1 0 1 0 0`
 ` -1/√2 i/√2 i/√2 -1/√2`
 ` 1/√2 -i/√2 -i/√2 1/√2`
 ` 0 0 e-i5π/12√2/2 e-i5π/12√2/2 0 0 eiπ/12√2/2 e-i11π/12√2/2 e-i7π/12√2/2 e-iπ/12√2/2 0 0 ei5π/12√2/2 e-iπ/12√2/2 0 0`
45
 ` 0 0 1 0 1 0 -1 0 0`
 ` -√2/2 √2/2 -√2/2 -√2/2`
d4+010
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 0 1 0 0 0 0 1`
 ` 0 1 0 -1 0 0 0 0 -1`
 ` -1/√2 1/√2 -1/√2 -1/√2`
 ` 1/√2 -1/√2 1/√2 1/√2`
 ` 0 0 ei5π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 ei11π/12√2/2 ei7π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 eiπ/12√2/2 0 0`
46
 ` 0 0 1 0 -1 0 1 0 0`
 ` i√2/2 i√2/2 i√2/2 -i√2/2`
d2101
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 -1 0 -1 0 0 0 0 1`
 ` i/√2 i/√2 i/√2 -i/√2`
 ` -i/√2 -i/√2 -i/√2 i/√2`
 ` 0 0 ei11π/12√2/2 ei5π/12√2/2 0 0 e-iπ/12√2/2 ei5π/12√2/2 eiπ/12√2/2 e-i11π/12√2/2 0 0 ei7π/12√2/2 ei7π/12√2/2 0 0`
47
 ` 0 0 -1 0 1 0 1 0 0`
 ` -√2/2 -√2/2 √2/2 -√2/2`
d4-010
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 0 -1 0 0 0 0 1`
 ` 0 -1 0 1 0 0 0 0 -1`
 ` -1/√2 -1/√2 1/√2 -1/√2`
 ` 1/√2 1/√2 -1/√2 1/√2`
 ` 0 0 e-i7π/12√2/2 ei11π/12√2/2 0 0 e-i7π/12√2/2 e-iπ/12√2/2 e-i5π/12√2/2 ei7π/12√2/2 0 0 e-i11π/12√2/2 e-i11π/12√2/2 0 0`
48
 ` 0 0 -1 0 -1 0 -1 0 0`
 ` -i√2/2 i√2/2 i√2/2 i√2/2`
d2-101
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 1 0 1 0 0 0 0 1`
 ` -i/√2 i/√2 i/√2 i/√2`
 ` i/√2 -i/√2 -i/√2 -i/√2`
 ` 0 0 ei11π/12√2/2 e-i7π/12√2/2 0 0 ei11π/12√2/2 ei5π/12√2/2 eiπ/12√2/2 eiπ/12√2/2 0 0 e-i5π/12√2/2 ei7π/12√2/2 0 0`
k-Subgroupsmag
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