Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group m'-3' (N. 29.3.111)


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
GM2GM3
1E2E
GM2GM3
2
2
-1
-1
2
-1
-1
GM4
T
GM4
3
-1
0
0
3
0
0
GM5
E
GM5
2
0
1
1
-2
-1
-1
GM7GM6
1F2F
GM6GM7
4
0
-1
-1
-4
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: 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 0
 0 1
)
(
 1 0 0
 0 1 0
 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 0
 0 1
)
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 -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 0
 0 1
)
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
(
  0 -1  0  0
  1  0  0  0
  0  0  0 -1
  0  0  1  0
)
4
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
2100
1
(
 1 0
 0 1
)
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
(
  0 -i  0  0
 -i  0  0  0
  0  0  0  i
  0  0  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  0
  0 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    0    0
   ei5π/12√2/2  ei11π/12√2/2    0    0
    0    0  e-i5π/12√2/2    eiπ/12√2/2
    0    0  e-i5π/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
(
  ei2π/3  0
  0 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    0    0
  ei11π/12√2/2   ei5π/12√2/2    0    0
    0    0 e-i11π/12√2/2   ei7π/12√2/2
    0    0 e-i11π/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
(
  ei2π/3  0
  0 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    0    0
   e-iπ/12√2/2   ei5π/12√2/2    0    0
    0    0 e-i11π/12√2/2  e-i5π/12√2/2
    0    0    eiπ/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
(
  ei2π/3  0
  0 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    0    0
  e-i7π/12√2/2  ei11π/12√2/2    0    0
    0    0  e-i5π/12√2/2 e-i11π/12√2/2
    0    0   ei7π/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
(
 e-i2π/3  0
  0  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    0    0
    eiπ/12√2/2 e-i11π/12√2/2    0    0
    0    0   ei5π/12√2/2   ei5π/12√2/2
    0    0   e-iπ/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
(
 e-i2π/3  0
  0  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    0    0
  e-i5π/12√2/2  e-i5π/12√2/2    0    0
    0    0  ei11π/12√2/2   e-iπ/12√2/2
    0    0   ei5π/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
(
 e-i2π/3  0
  0  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    0    0
 e-i11π/12√2/2 e-i11π/12√2/2    0    0
    0    0   ei5π/12√2/2  e-i7π/12√2/2
    0    0  ei11π/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
(
 e-i2π/3  0
  0  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    0    0
   ei7π/12√2/2  e-i5π/12√2/2    0    0
    0    0  ei11π/12√2/2  ei11π/12√2/2
    0    0  e-i7π/12√2/2   ei5π/12√2/2
)
13
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
(
 1 0
 0 1
)
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0  0  0
  0 -1  0  0
  0  0 -1  0
  0  0  0 -1
)
14
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
(
 1 0
 0 1
)
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  i  0
  0 -i
)
(
  i  0  0  0
  0 -i  0  0
  0  0 -i  0
  0  0  0  i
)
15
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2010
1
(
 1 0
 0 1
)
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
(
  0  1  0  0
 -1  0  0  0
  0  0  0  1
  0  0 -1  0
)
16
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
d2100
1
(
 1 0
 0 1
)
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
 0 i
 i 0
)
(
  0  i  0  0
  i  0  0  0
  0  0  0 -i
  0  0 -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  0
  0 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    0    0
  e-i7π/12√2/2   e-iπ/12√2/2    0    0
    0    0   ei7π/12√2/2 e-i11π/12√2/2
    0    0   ei7π/12√2/2    eiπ/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  0
  0 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    0    0
  e-iπ/12√2/2 e-i7π/12√2/2    0    0
    0    0   eiπ/12√2/2 e-i5π/12√2/2
    0    0   eiπ/12√2/2  ei7π/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  0
  0 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    0    0
  ei11π/12√2/2  e-i7π/12√2/2    0    0
    0    0    eiπ/12√2/2   ei7π/12√2/2
    0    0 e-i11π/12√2/2   ei7π/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  0
  0 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    0    0
  ei5π/12√2/2  e-iπ/12√2/2    0    0
    0    0  ei7π/12√2/2   eiπ/12√2/2
    0    0 e-i5π/12√2/2   eiπ/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  0
  0  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    0    0
 e-i11π/12√2/2    eiπ/12√2/2    0    0
    0    0  e-i7π/12√2/2  e-i7π/12√2/2
    0    0  ei11π/12√2/2   e-iπ/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  0
  0  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    0    0
   ei7π/12√2/2   ei7π/12√2/2    0    0
    0    0   e-iπ/12√2/2  ei11π/12√2/2
    0    0  e-i7π/12√2/2  e-i7π/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  0
  0  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    0    0
   eiπ/12√2/2   eiπ/12√2/2    0    0
    0    0 e-i7π/12√2/2  ei5π/12√2/2
    0    0  e-iπ/12√2/2  e-iπ/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  0
  0  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    0    0
 e-i5π/12√2/2  ei7π/12√2/2    0    0
    0    0  e-iπ/12√2/2  e-iπ/12√2/2
    0    0  ei5π/12√2/2 e-i7π/12√2/2
)
25
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1'
1
(
 0 1
 1 0
)
(
 1 0 0
 0 1 0
 0 0 1
)
(
  0  1
 -1  0
)
(
  0  0 -1  0
  0  0  0 -1
  1  0  0  0
  0  1  0  0
)
26
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m'001
1
(
 0 1
 1 0
)
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
(
  0  0  i  0
  0  0  0 -i
  i  0  0  0
  0 -i  0  0
)
27
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
m'010
1
(
 0 1
 1 0
)
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 1 0
 0 1
)
(
  0  0  0  1
  0  0 -1  0
  0 -1  0  0
  1  0  0  0
)
28
(
 -1  0  0
  0  1  0
  0  0  1
)
(
  0 -i
 -i  0
)
m'100
1
(
 0 1
 1 0
)
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
(
 0 0 0 i
 0 0 i 0
 0 i 0 0
 i 0 0 0
)
29
(
  0  0 -1
 -1  0  0
  0 -1  0
)
(
  (1-i)/2 -(1+i)/2
  (1-i)/2  (1+i)/2
)
3'+111
1
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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    eiπ/12√2/2    0    0
  e-i5π/12√2/2 e-i11π/12√2/2    0    0
)
30
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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   ei7π/12√2/2    0    0
 e-i11π/12√2/2  e-i5π/12√2/2    0    0
)
31
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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  e-i5π/12√2/2    0    0
    eiπ/12√2/2  e-i5π/12√2/2    0    0
)
32
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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-i11π/12√2/2    0    0
   ei7π/12√2/2 e-i11π/12√2/2    0    0
)
33
(
  0 -1  0
  0  0 -1
 -1  0  0
)
(
  (1+i)/2  (1+i)/2
 -(1-i)/2  (1-i)/2
)
3'-111
1
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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   ei5π/12√2/2    0    0
   e-iπ/12√2/2  ei11π/12√2/2    0    0
)
34
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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   e-iπ/12√2/2    0    0
   ei5π/12√2/2   ei5π/12√2/2    0    0
)
35
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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-i7π/12√2/2    0    0
 ei11π/12√2/2 ei11π/12√2/2    0    0
)
36
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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 ei11π/12√2/2    0    0
 e-i7π/12√2/2  ei5π/12√2/2    0    0
)
37
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1'
1
(
 0 1
 1 0
)
(
 1 0 0
 0 1 0
 0 0 1
)
(
  0 -1
  1  0
)
(
  0  0  1  0
  0  0  0  1
 -1  0  0  0
  0 -1  0  0
)
38
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm'001
1
(
 0 1
 1 0
)
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
(
  0  0 -i  0
  0  0  0  i
 -i  0  0  0
  0  i  0  0
)
39
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
dm'010
1
(
 0 1
 1 0
)
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -1  0
  0 -1
)
(
  0  0  0 -1
  0  0  1  0
  0  1  0  0
 -1  0  0  0
)
40
(
 -1  0  0
  0  1  0
  0  0  1
)
(
 0 i
 i 0
)
dm'100
1
(
 0 1
 1 0
)
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
(
  0  0  0 -i
  0  0 -i  0
  0 -i  0  0
 -i  0  0  0
)
41
(
  0  0 -1
 -1  0  0
  0 -1  0
)
(
 -(1-i)/2  (1+i)/2
 -(1-i)/2 -(1+i)/2
)
d3'+111
1
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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-i11π/12√2/2    0    0
   ei7π/12√2/2    eiπ/12√2/2    0    0
)
42
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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 e-i5π/12√2/2    0    0
   eiπ/12√2/2  ei7π/12√2/2    0    0
)
43
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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   ei7π/12√2/2    0    0
 e-i11π/12√2/2   ei7π/12√2/2    0    0
)
44
(
  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
(
  0  ei2π/3
 e-i2π/3  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
)
(
    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   eiπ/12√2/2    0    0
 e-i5π/12√2/2   eiπ/12√2/2    0    0
)
45
(
  0 -1  0
  0  0 -1
 -1  0  0
)
(
 -(1+i)/2 -(1+i)/2
  (1-i)/2 -(1-i)/2
)
d3'-111
1
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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-i7π/12√2/2    0    0
  ei11π/12√2/2   e-iπ/12√2/2    0    0
)
46
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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  ei11π/12√2/2    0    0
  e-i7π/12√2/2  e-i7π/12√2/2    0    0
)
47
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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   ei5π/12√2/2    0    0
   e-iπ/12√2/2   e-iπ/12√2/2    0    0
)
48
(
  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
(
  0 e-i2π/3
  ei2π/3  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
)
(
    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   e-iπ/12√2/2    0    0
   ei5π/12√2/2  e-i7π/12√2/2    0    0
)
k-Subgroupsmag
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