Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group m-3m' (N. 32.4.121)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
-1
-1
-1
-1
1
1
1
-1
-1
-1
GM2+
1Eg
GM2+
1
1
-(1-i√3)/2
-(1+i√3)/2
1
1
-(1-i√3)/2
-(1+i√3)/2
1
-(1-i√3)/2
-(1+i√3)/2
1
-(1-i√3)/2
-(1+i√3)/2
GM2-
1Eu
GM2-
1
1
-(1-i√3)/2
-(1+i√3)/2
-1
-1
(1-i√3)/2
(1+i√3)/2
1
-(1-i√3)/2
-(1+i√3)/2
-1
(1-i√3)/2
(1+i√3)/2
GM3+
2Eg
GM3+
1
1
-(1+i√3)/2
-(1-i√3)/2
1
1
-(1+i√3)/2
-(1-i√3)/2
1
-(1+i√3)/2
-(1-i√3)/2
1
-(1+i√3)/2
-(1-i√3)/2
GM3-
2Eu
GM3-
1
1
-(1+i√3)/2
-(1-i√3)/2
-1
-1
(1+i√3)/2
(1-i√3)/2
1
-(1+i√3)/2
-(1-i√3)/2
-1
(1+i√3)/2
(1-i√3)/2
GM4+
Tg
GM4+
3
-1
0
0
3
-1
0
0
3
0
0
3
0
0
GM4-
Tu
GM4-
3
-1
0
0
-3
1
0
0
3
0
0
-3
0
0
GM5+
Eg
GM5
2
0
1
1
2
0
1
1
-2
-1
-1
-2
-1
-1
GM7+
2Fg
GM6
2
0
-(1-i√3)/2
-(1+i√3)/2
2
0
-(1-i√3)/2
-(1+i√3)/2
-2
(1-i√3)/2
(1+i√3)/2
-2
(1-i√3)/2
(1+i√3)/2
GM6+
1Fg
GM7
2
0
-(1+i√3)/2
-(1-i√3)/2
2
0
-(1+i√3)/2
-(1-i√3)/2
-2
(1+i√3)/2
(1-i√3)/2
-2
(1+i√3)/2
(1-i√3)/2
GM5-
Eu
GM8
2
0
1
1
-2
0
-1
-1
-2
-1
-1
2
1
1
GM7-
2Fu
GM9
2
0
-(1-i√3)/2
-(1+i√3)/2
-2
0
(1-i√3)/2
(1+i√3)/2
-2
(1-i√3)/2
(1+i√3)/2
2
-(1-i√3)/2
-(1+i√3)/2
GM6-
1Fu
GM10
2
0
-(1+i√3)/2
-(1-i√3)/2
-2
0
(1+i√3)/2
(1-i√3)/2
-2
(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: 1
C6: m001, m010, m100, dm001, dm010, dm100
C7: 3+111, 3+111, 3+111, 3+111
C8: 3-111, 3-111, 3-111, 3-111
C9: d1
C10: d3+111, d3+111, d3+111, d3+111
C11: d3-111, d3-111, d3-111, d3-111
C12: d1
C13: d3+111, d3+111, d3+111, d3+111
C14: 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 symbolGM1+GM1-GM2+GM2-GM3+GM3-GM4+GM4-GM5GM6GM7GM8GM9GM10
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
1
1
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 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
1
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  i
)
(
 -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
1
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
  1  0
)
(
  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
1
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
 -i  0
)
(
  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
1
ei2π/3
ei2π/3
e-i2π/3
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
)
(
  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
)
(
  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
1
ei2π/3
ei2π/3
e-i2π/3
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
)
(
 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
)
(
  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
1
ei2π/3
ei2π/3
e-i2π/3
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
)
(
 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
)
(
  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
1
ei2π/3
ei2π/3
e-i2π/3
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
)
(
  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
)
(
 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
1
e-i2π/3
e-i2π/3
ei2π/3
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
)
(
  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
)
(
  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
1
e-i2π/3
e-i2π/3
ei2π/3
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-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
)
(
 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
1
e-i2π/3
e-i2π/3
ei2π/3
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
)
(
  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
)
(
  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
1
e-i2π/3
e-i2π/3
ei2π/3
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-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
)
(
  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
)
1
1
-1
1
-1
1
-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 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
)
m001
1
-1
1
-1
1
-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  i
)
(
  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
)
m010
1
-1
1
-1
1
-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
  1  0
)
(
  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
)
m100
1
-1
1
-1
1
-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
 -i  0
)
(
 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
)
3+111
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
  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
)
(
  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
)
3+-11-1
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
 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
)
(
 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
)
3+1-1-1
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
 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
)
(
 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
)
3+-1-11
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
  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
)
(
  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
)
3-111
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
 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
)
3-1-1-1
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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-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
)
(
  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
)
3--1-11
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
 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
)
3--11-1
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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-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
)
(
  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
)
25
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
1
1
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 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
26
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
1
1
1
1
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 -i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
27
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2010
1
1
1
1
1
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
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
28
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
d2100
1
1
1
1
1
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
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 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
ei2π/3
e-i2π/3
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
)
(
 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
)
(
  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
)
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
ei2π/3
e-i2π/3
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-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
)
(
 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
)
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
ei2π/3
e-i2π/3
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-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
)
(
 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
)
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
ei2π/3
e-i2π/3
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
)
(
 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
)
(
  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
)
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
e-i2π/3
ei2π/3
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
)
(
  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
)
(
 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
)
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
e-i2π/3
ei2π/3
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
)
(
  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
)
(
  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
)
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
e-i2π/3
ei2π/3
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
)
(
  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
)
(
 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
)
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
e-i2π/3
ei2π/3
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
)
(
  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
)
(
  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
)
37
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1
1
-1
1
-1
1
-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 -1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
38
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm001
1
-1
1
-1
1
-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 -i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
39
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
dm010
1
-1
1
-1
1
-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
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
40
(
 -1  0  0
  0  1  0
  0  0  1
)
(
 0 i
 i 0
)
dm100
1
-1
1
-1
1
-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
 i 0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  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
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
 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
)
(
  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
)
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
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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-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
)
(
  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
)
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
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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-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
)
(
  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
)
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
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/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
)
(
 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
)
(
 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
)
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
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
  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
)
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
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
 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
)
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
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
  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
)
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
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/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
)
(
  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
)
(
  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
)
49
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
2'110
1
1
1
1
1
1
(
 -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
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
50
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
2'1-10
1
1
1
1
1
1
(
 -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
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
51
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4'-001
1
1
1
1
1
1
(
  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 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
52
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4'+001
1
1
1
1
1
1
(
  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 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
53
(
  1  0  0
  0  0  1
  0 -1  0
)
(
  √2/2 i√2/2
 i√2/2  √2/2
)
4'-100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  e-iπ/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2  eiπ/6√2/2
)
(
  i/√2 -1/√2
  1/√2 -i/√2
)
(
  e-iπ/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2  eiπ/6√2/2
)
54
(
 -1  0  0
  0  0  1
  0  1  0
)
(
  i√2/2  √2/2
  -√2/2 -i√2/2
)
2'011
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
 e-i2π/3√2/2  ei5π/6√2/2
  ei5π/6√2/2 e-i2π/3√2/2
)
(
 ei2π/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 ei2π/3√2/2
)
(
  1/√2 -i/√2
 -i/√2  1/√2
)
(
 e-i2π/3√2/2  ei5π/6√2/2
  ei5π/6√2/2 e-i2π/3√2/2
)
(
 ei2π/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 ei2π/3√2/2
)
55
(
 -1  0  0
  0  0 -1
  0 -1  0
)
(
 -i√2/2  √2/2
  -√2/2  i√2/2
)
2'01-1
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
 e-i2π/3√2/2  e-iπ/6√2/2
  e-iπ/6√2/2 e-i2π/3√2/2
)
(
  ei2π/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  ei2π/3√2/2
)
(
 1/√2 i/√2
 i/√2 1/√2
)
(
 e-i2π/3√2/2  e-iπ/6√2/2
  e-iπ/6√2/2 e-i2π/3√2/2
)
(
  ei2π/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  ei2π/3√2/2
)
56
(
  1  0  0
  0  0 -1
  0  1  0
)
(
  √2/2 -i√2/2
 -i√2/2  √2/2
)
4'+100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  ei5π/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2 e-i5π/6√2/2
)
(
 -i/√2 -1/√2
  1/√2  i/√2
)
(
  ei5π/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2 e-i5π/6√2/2
)
57
(
  0  0  1
  0  1  0
 -1  0  0
)
(
  √2/2 -√2/2
  √2/2  √2/2
)
4'+010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 e-iπ/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2  eiπ/3√2/2
)
(
 -1/√2 -1/√2
  1/√2 -1/√2
)
(
 e-iπ/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2  eiπ/3√2/2
)
58
(
  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
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
  eiπ/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 e-iπ/6√2/2
)
(
 -i/√2  i/√2
  i/√2  i/√2
)
(
  eiπ/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 e-iπ/6√2/2
)
59
(
  0  0 -1
  0  1  0
  1  0  0
)
(
  √2/2  √2/2
 -√2/2  √2/2
)
4'-010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 ei2π/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2 e-i2π/3√2/2
)
(
  1/√2 -1/√2
  1/√2  1/√2
)
(
 ei2π/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2 e-i2π/3√2/2
)
60
(
  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
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
  eiπ/6√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 e-iπ/6√2/2
)
(
 -i/√2 -i/√2
 -i/√2  i/√2
)
(
  eiπ/6√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 e-iπ/6√2/2
)
61
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
m'110
1
-1
1
-1
1
-1
(
 -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
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
62
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
m'1-10
1
-1
1
-1
1
-1
(
 -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
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
63
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4'-001
1
-1
1
-1
1
-1
(
  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 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
64
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4'+001
1
-1
1
-1
1
-1
(
  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 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
65
(
 -1  0  0
  0  0 -1
  0  1  0
)
(
  √2/2 i√2/2
 i√2/2  √2/2
)
4'-100
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  e-iπ/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2  eiπ/6√2/2
)
(
 -i/√2  1/√2
 -1/√2  i/√2
)
(
  ei5π/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2 e-i5π/6√2/2
)
66
(
  1  0  0
  0  0 -1
  0 -1  0
)
(
  i√2/2  √2/2
  -√2/2 -i√2/2
)
m'011
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
 e-i2π/3√2/2  ei5π/6√2/2
  ei5π/6√2/2 e-i2π/3√2/2
)
(
 ei2π/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 ei2π/3√2/2
)
(
 -1/√2  i/√2
  i/√2 -1/√2
)
(
  eiπ/3√2/2 e-iπ/6√2/2
 e-iπ/6√2/2  eiπ/3√2/2
)
(
  e-iπ/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  e-iπ/3√2/2
)
67
(
 1 0 0
 0 0 1
 0 1 0
)
(
 -i√2/2  √2/2
  -√2/2  i√2/2
)
m'01-1
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
 e-i2π/3√2/2  e-iπ/6√2/2
  e-iπ/6√2/2 e-i2π/3√2/2
)
(
  ei2π/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  ei2π/3√2/2
)
(
 -1/√2 -i/√2
 -i/√2 -1/√2
)
(
  eiπ/3√2/2 ei5π/6√2/2
 ei5π/6√2/2  eiπ/3√2/2
)
(
 e-iπ/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-iπ/3√2/2
)
68
(
 -1  0  0
  0  0  1
  0 -1  0
)
(
  √2/2 -i√2/2
 -i√2/2  √2/2
)
4'+100
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  ei5π/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2 e-i5π/6√2/2
)
(
  i/√2  1/√2
 -1/√2 -i/√2
)
(
  e-iπ/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2  eiπ/6√2/2
)
69
(
  0  0 -1
  0 -1  0
  1  0  0
)
(
  √2/2 -√2/2
  √2/2  √2/2
)
4'+010
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 e-iπ/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2  eiπ/3√2/2
)
(
  1/√2  1/√2
 -1/√2  1/√2
)
(
 ei2π/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2 e-i2π/3√2/2
)
70
(
  0  0 -1
  0  1  0
 -1  0  0
)
(
 -i√2/2 -i√2/2
 -i√2/2  i√2/2
)
m'101
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
  eiπ/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 e-iπ/6√2/2
)
(
  i/√2 -i/√2
 -i/√2 -i/√2
)
(
 e-i5π/6√2/2  eiπ/6√2/2
  eiπ/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 ei5π/6√2/2
)
71
(
  0  0  1
  0 -1  0
 -1  0  0
)
(
  √2/2  √2/2
 -√2/2  √2/2
)
4'-010
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 ei2π/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2 e-i2π/3√2/2
)
(
 -1/√2  1/√2
 -1/√2 -1/√2
)
(
 e-iπ/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2  eiπ/3√2/2
)
72
(
 0 0 1
 0 1 0
 1 0 0
)
(
  i√2/2 -i√2/2
 -i√2/2 -i√2/2
)
m'-101
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
  eiπ/6√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 e-iπ/6√2/2
)
(
  i/√2  i/√2
  i/√2 -i/√2
)
(
 e-i5π/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 ei5π/6√2/2
)
73
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
d2'110
1
1
1
1
1
1
(
 -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
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
74
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
d2'1-10
1
1
1
1
1
1
(
 -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
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
75
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4'-001
1
1
1
1
1
1
(
  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  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
76
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4'+001
1
1
1
1
1
1
(
  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  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
77
(
  1  0  0
  0  0  1
  0 -1  0
)
(
  -√2/2 -i√2/2
 -i√2/2  -√2/2
)
d4'-100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  ei5π/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2 e-i5π/6√2/2
)
(
 -i/√2  1/√2
 -1/√2  i/√2
)
(
  ei5π/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2 e-i5π/6√2/2
)
78
(
 -1  0  0
  0  0  1
  0  1  0
)
(
 -i√2/2  -√2/2
  √2/2  i√2/2
)
d2'011
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  eiπ/3√2/2 e-iπ/6√2/2
 e-iπ/6√2/2  eiπ/3√2/2
)
(
  e-iπ/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  e-iπ/3√2/2
)
(
 -1/√2  i/√2
  i/√2 -1/√2
)
(
  eiπ/3√2/2 e-iπ/6√2/2
 e-iπ/6√2/2  eiπ/3√2/2
)
(
  e-iπ/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  e-iπ/3√2/2
)
79
(
 -1  0  0
  0  0 -1
  0 -1  0
)
(
  i√2/2  -√2/2
  √2/2 -i√2/2
)
d2'01-1
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  eiπ/3√2/2 ei5π/6√2/2
 ei5π/6√2/2  eiπ/3√2/2
)
(
 e-iπ/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-iπ/3√2/2
)
(
 -1/√2 -i/√2
 -i/√2 -1/√2
)
(
  eiπ/3√2/2 ei5π/6√2/2
 ei5π/6√2/2  eiπ/3√2/2
)
(
 e-iπ/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-iπ/3√2/2
)
80
(
  1  0  0
  0  0 -1
  0  1  0
)
(
 -√2/2 i√2/2
 i√2/2 -√2/2
)
d4'+100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
  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
)
(
  e-iπ/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2  eiπ/6√2/2
)
(
  i/√2  1/√2
 -1/√2 -i/√2
)
(
  e-iπ/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2  eiπ/6√2/2
)
81
(
  0  0  1
  0  1  0
 -1  0  0
)
(
 -√2/2  √2/2
 -√2/2 -√2/2
)
d4'+010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 ei2π/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2 e-i2π/3√2/2
)
(
  1/√2  1/√2
 -1/√2  1/√2
)
(
 ei2π/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2 e-i2π/3√2/2
)
82
(
  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
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 e-i5π/6√2/2  eiπ/6√2/2
  eiπ/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 ei5π/6√2/2
)
(
  i/√2 -i/√2
 -i/√2 -i/√2
)
(
 e-i5π/6√2/2  eiπ/6√2/2
  eiπ/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 ei5π/6√2/2
)
83
(
  0  0 -1
  0  1  0
  1  0  0
)
(
 -√2/2 -√2/2
  √2/2 -√2/2
)
d4'-010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 e-iπ/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2  eiπ/3√2/2
)
(
 -1/√2  1/√2
 -1/√2 -1/√2
)
(
 e-iπ/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2  eiπ/3√2/2
)
84
(
  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
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
  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
)
(
 e-i5π/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 ei5π/6√2/2
)
(
  i/√2  i/√2
  i/√2 -i/√2
)
(
 e-i5π/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 ei5π/6√2/2
)
85
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
dm'110
1
-1
1
-1
1
-1
(
 -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
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
86
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
dm'1-10
1
-1
1
-1
1
-1
(
 -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
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
87
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4'-001
1
-1
1
-1
1
-1
(
  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  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
88
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4'+001
1
-1
1
-1
1
-1
(
  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  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
89
(
 -1  0  0
  0  0 -1
  0  1  0
)
(
  -√2/2 -i√2/2
 -i√2/2  -√2/2
)
d4'-100
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  ei5π/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2 e-i5π/6√2/2
)
(
  i/√2 -1/√2
  1/√2 -i/√2
)
(
  e-iπ/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2  eiπ/6√2/2
)
90
(
  1  0  0
  0  0 -1
  0 -1  0
)
(
 -i√2/2  -√2/2
  √2/2  i√2/2
)
dm'011
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  eiπ/3√2/2 e-iπ/6√2/2
 e-iπ/6√2/2  eiπ/3√2/2
)
(
  e-iπ/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  e-iπ/3√2/2
)
(
  1/√2 -i/√2
 -i/√2  1/√2
)
(
 e-i2π/3√2/2  ei5π/6√2/2
  ei5π/6√2/2 e-i2π/3√2/2
)
(
 ei2π/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 ei2π/3√2/2
)
91
(
 1 0 0
 0 0 1
 0 1 0
)
(
  i√2/2  -√2/2
  √2/2 -i√2/2
)
dm'01-1
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  eiπ/3√2/2 ei5π/6√2/2
 ei5π/6√2/2  eiπ/3√2/2
)
(
 e-iπ/3√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-iπ/3√2/2
)
(
 1/√2 i/√2
 i/√2 1/√2
)
(
 e-i2π/3√2/2  e-iπ/6√2/2
  e-iπ/6√2/2 e-i2π/3√2/2
)
(
  ei2π/3√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  ei2π/3√2/2
)
92
(
 -1  0  0
  0  0  1
  0 -1  0
)
(
 -√2/2 i√2/2
 i√2/2 -√2/2
)
d4'+100
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
  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
)
(
  e-iπ/6√2/2 e-i2π/3√2/2
  eiπ/3√2/2  ei5π/6√2/2
)
(
 e-i5π/6√2/2  ei2π/3√2/2
  e-iπ/3√2/2  eiπ/6√2/2
)
(
 -i/√2 -1/√2
  1/√2  i/√2
)
(
  ei5π/6√2/2  eiπ/3√2/2
 e-i2π/3√2/2  e-iπ/6√2/2
)
(
  eiπ/6√2/2  e-iπ/3√2/2
  ei2π/3√2/2 e-i5π/6√2/2
)
93
(
  0  0 -1
  0 -1  0
  1  0  0
)
(
 -√2/2  √2/2
 -√2/2 -√2/2
)
d4'+010
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 ei2π/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2 e-i2π/3√2/2
)
(
 -1/√2 -1/√2
  1/√2 -1/√2
)
(
 e-iπ/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2  eiπ/3√2/2
)
94
(
  0  0 -1
  0  1  0
 -1  0  0
)
(
  i√2/2  i√2/2
  i√2/2 -i√2/2
)
dm'101
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 e-i5π/6√2/2  eiπ/6√2/2
  eiπ/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 ei5π/6√2/2
)
(
 -i/√2  i/√2
  i/√2  i/√2
)
(
  eiπ/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 e-iπ/6√2/2
)
95
(
  0  0  1
  0 -1  0
 -1  0  0
)
(
 -√2/2 -√2/2
  √2/2 -√2/2
)
d4'-010
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 e-iπ/3√2/2 ei2π/3√2/2
 e-iπ/3√2/2 e-iπ/3√2/2
)
(
  eiπ/3√2/2 e-i2π/3√2/2
  eiπ/3√2/2  eiπ/3√2/2
)
(
  1/√2 -1/√2
  1/√2  1/√2
)
(
 ei2π/3√2/2 e-iπ/3√2/2
 ei2π/3√2/2 ei2π/3√2/2
)
(
 e-i2π/3√2/2  eiπ/3√2/2
 e-i2π/3√2/2 e-i2π/3√2/2
)
96
(
 0 0 1
 0 1 0
 1 0 0
)
(
 -i√2/2  i√2/2
  i√2/2  i√2/2
)
dm'-101
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
  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
)
(
 e-i5π/6√2/2 e-i5π/6√2/2
 e-i5π/6√2/2  eiπ/6√2/2
)
(
 e-iπ/6√2/2 e-iπ/6√2/2
 e-iπ/6√2/2 ei5π/6√2/2
)
(
 -i/√2 -i/√2
 -i/√2  i/√2
)
(
  eiπ/6√2/2  eiπ/6√2/2
  eiπ/6√2/2 e-i5π/6√2/2
)
(
 ei5π/6√2/2 ei5π/6√2/2
 ei5π/6√2/2 e-iπ/6√2/2
)
k-Subgroupsmag
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