Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 4321' (N. 30.2.113)


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, 2100d2001d2010d2100
C3: 3+111, 3+111, 3+111, 3+111, 3-111, 3-111, 3-111, 3-111
C4: 2110, 2110, 2011, 2011, 2101, 2101d2110d2110d2011d2011d2101d2101
C5: 4-001, 4+001, 4-100, 4+100, 4+010, 4-010
C6d1
C7d3+111d3+111d3+111d3+111d3-111d3-111d3-111d3-111
C8d4-001d4+001d4-100d4+100d4+010d4-010

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/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
e-iπ/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
ei5π/122/2 e-iπ/122/2 0 0
ei5π/122/2 ei11π/122/2 0 0
0 0 e-i5π/122/2 ei7π/122/2
0 0 e-i11π/122/2 e-i11π/122/2
)
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/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
eiπ/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
ei11π/122/2 e-i7π/122/2 0 0
ei11π/122/2 ei5π/122/2 0 0
0 0 e-i11π/122/2 e-i11π/122/2
0 0 ei7π/122/2 e-i5π/122/2
)
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/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
eiπ/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
ei11π/122/2 ei5π/122/2 0 0
e-iπ/122/2 ei5π/122/2 0 0
0 0 e-i11π/122/2 eiπ/122/2
0 0 e-i5π/122/2 e-i5π/122/2
)
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/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
e-iπ/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
ei5π/122/2 ei11π/122/2 0 0
e-i7π/122/2 ei11π/122/2 0 0
0 0 e-i5π/122/2 e-i5π/122/2
0 0 eiπ/122/2 e-i11π/122/2
)
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/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
eiπ/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
e-i5π/122/2 e-i5π/122/2 0 0
eiπ/122/2 e-i11π/122/2 0 0
0 0 ei5π/122/2 ei11π/122/2
0 0 e-i7π/122/2 ei11π/122/2
)
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/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
e-iπ/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
e-i11π/122/2 eiπ/122/2 0 0
e-i5π/122/2 e-i5π/122/2 0 0
0 0 ei11π/122/2 ei5π/122/2
0 0 e-iπ/122/2 ei5π/122/2
)
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/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
eiπ/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
e-i5π/122/2 ei7π/122/2 0 0
e-i11π/122/2 e-i11π/122/2 0 0
0 0 ei5π/122/2 e-iπ/122/2
0 0 ei5π/122/2 ei11π/122/2
)
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/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
e-iπ/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
e-i11π/122/2 e-i11π/122/2 0 0
ei7π/122/2 e-i5π/122/2 0 0
0 0 ei11π/122/2 e-i7π/122/2
0 0 ei11π/122/2 ei5π/122/2
)
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 i2/2
i2/2 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/122/2 ei7π/122/2
0 0 e-i11π/122/2 e-i11π/122/2
e-i7π/122/2 ei11π/122/2 0 0
e-i7π/122/2 e-iπ/122/2 0 0
)
18
(
-1 0 0
0 0 1
0 1 0
)
(
i2/2 2/2
-2/2 -i2/2
)
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/122/2 e-i11π/122/2
0 0 ei7π/122/2 e-i5π/122/2
e-iπ/122/2 ei5π/122/2 0 0
e-iπ/122/2 e-i7π/122/2 0 0
)
19
(
-1 0 0
0 0 -1
0 -1 0
)
(
-i2/2 2/2
-2/2 i2/2
)
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/122/2 eiπ/122/2
0 0 e-i5π/122/2 e-i5π/122/2
e-iπ/122/2 e-i7π/122/2 0 0
ei11π/122/2 e-i7π/122/2 0 0
)
20
(
1 0 0
0 0 -1
0 1 0
)
(
2/2 -i2/2
-i2/2 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/122/2 ei7π/122/2
0 0 e-i11π/122/2 eiπ/122/2
ei5π/122/2 ei11π/122/2 0 0
e-i7π/122/2 ei11π/122/2 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/122/2 e-iπ/122/2
0 0 ei5π/122/2 e-iπ/122/2
e-i5π/122/2 e-i5π/122/2 0 0
eiπ/122/2 e-i11π/122/2 0 0
)
22
(
0 0 1
0 -1 0
1 0 0
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
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/122/2 e-i7π/122/2
0 0 ei11π/122/2 e-i7π/122/2
e-i11π/122/2 eiπ/122/2 0 0
e-i5π/122/2 e-i5π/122/2 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/122/2 e-iπ/122/2
0 0 ei5π/122/2 ei11π/122/2
ei7π/122/2 e-i5π/122/2 0 0
eiπ/122/2 eiπ/122/2 0 0
)
24
(
0 0 -1
0 -1 0
-1 0 0
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
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/122/2 ei5π/122/2
0 0 e-iπ/122/2 e-i7π/122/2
e-i11π/122/2 e-i11π/122/2 0 0
ei7π/122/2 e-i5π/122/2 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/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
ei3π/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
e-i7π/122/2 ei11π/122/2 0 0
e-i7π/122/2 e-iπ/122/2 0 0
0 0 ei7π/122/2 e-i5π/122/2
0 0 eiπ/122/2 eiπ/122/2
)
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/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
e-i3π/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
e-iπ/122/2 ei5π/122/2 0 0
e-iπ/122/2 e-i7π/122/2 0 0
0 0 eiπ/122/2 eiπ/122/2
0 0 e-i5π/122/2 ei7π/122/2
)
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/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
e-i3π/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
e-iπ/122/2 e-i7π/122/2 0 0
ei11π/122/2 e-i7π/122/2 0 0
0 0 eiπ/122/2 e-i11π/122/2
0 0 ei7π/122/2 ei7π/122/2
)
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/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
ei3π/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
e-i7π/122/2 e-iπ/122/2 0 0
ei5π/122/2 e-iπ/122/2 0 0
0 0 ei7π/122/2 ei7π/122/2
0 0 e-i11π/122/2 eiπ/122/2
)
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/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
e-i3π/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
ei7π/122/2 ei7π/122/2 0 0
e-i11π/122/2 eiπ/122/2 0 0
0 0 e-i7π/122/2 e-iπ/122/2
0 0 ei5π/122/2 e-iπ/122/2
)
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/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
ei3π/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
eiπ/122/2 e-i11π/122/2 0 0
ei7π/122/2 ei7π/122/2 0 0
0 0 e-iπ/122/2 e-i7π/122/2
0 0 ei11π/122/2 e-i7π/122/2
)
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/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
e-i3π/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
ei7π/122/2 e-i5π/122/2 0 0
eiπ/122/2 eiπ/122/2 0 0
0 0 e-i7π/122/2 ei11π/122/2
0 0 e-i7π/122/2 e-iπ/122/2
)
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/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
ei3π/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
eiπ/122/2 eiπ/122/2 0 0
e-i5π/122/2 ei7π/122/2 0 0
0 0 e-iπ/122/2 ei5π/122/2
0 0 e-iπ/122/2 e-i7π/122/2
)
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 -i2/2
-i2/2 -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/122/2 e-i5π/122/2
0 0 eiπ/122/2 eiπ/122/2
ei5π/122/2 e-iπ/122/2 0 0
ei5π/122/2 ei11π/122/2 0 0
)
42
(
-1 0 0
0 0 1
0 1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
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/122/2 eiπ/122/2
0 0 e-i5π/122/2 ei7π/122/2
ei11π/122/2 e-i7π/122/2 0 0
ei11π/122/2 ei5π/122/2 0 0
)
43
(
-1 0 0
0 0 -1
0 -1 0
)
(
i2/2 -2/2
2/2 -i2/2
)
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/122/2 e-i11π/122/2
0 0 ei7π/122/2 ei7π/122/2
ei11π/122/2 ei5π/122/2 0 0
e-iπ/122/2 ei5π/122/2 0 0
)
44
(
1 0 0
0 0 -1
0 1 0
)
(
-2/2 i2/2
i2/2 -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/122/2 e-i5π/122/2
0 0 eiπ/122/2 e-i11π/122/2
e-i7π/122/2 e-iπ/122/2 0 0
ei5π/122/2 e-iπ/122/2 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/122/2 ei11π/122/2
0 0 e-i7π/122/2 ei11π/122/2
ei7π/122/2 ei7π/122/2 0 0
e-i11π/122/2 eiπ/122/2 0 0
)
46
(
0 0 1
0 -1 0
1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
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/122/2 ei5π/122/2
0 0 e-iπ/122/2 ei5π/122/2
eiπ/122/2 e-i11π/122/2 0 0
ei7π/122/2 ei7π/122/2 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/122/2 ei11π/122/2
0 0 e-i7π/122/2 e-iπ/122/2
e-i5π/122/2 ei7π/122/2 0 0
e-i11π/122/2 e-i11π/122/2 0 0
)
48
(
0 0 -1
0 -1 0
-1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
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/122/2 e-i7π/122/2
0 0 ei11π/122/2 ei5π/122/2
eiπ/122/2 eiπ/122/2 0 0
e-i5π/122/2 ei7π/122/2 0 0
)
49
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1'
1
1
(
0 -i
-i 0
)
(
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 0 e-i3π/4 0
0 0 0 ei3π/4
eiπ/4 0 0 0
0 e-iπ/4 0 0
)
50
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2'001
1
1
(
0 -i
-i 0
)
(
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 0 ei3π/4 0
0 0 0 e-i3π/4
ei3π/4 0 0 0
0 e-i3π/4 0 0
)
51
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2'010
1
1
(
0 -i
-i 0
)
(
-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
)
(
0 0 0 e-iπ/4
0 0 e-i3π/4 0
0 e-i3π/4 0 0
e-iπ/4 0 0 0
)
52
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2'100
1
1
(
0 -i
-i 0
)
(
-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
)
(
0 0 0 eiπ/4
0 0 ei3π/4 0
0 ei3π/4 0 0
eiπ/4 0 0 0
)
53
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
(
eiπ/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
eiπ/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
0 0 e-iπ/32/2 ei2π/32/2
0 0 e-iπ/32/2 e-iπ/32/2
e-iπ/62/2 eiπ/32/2 0 0
e-i2π/32/2 ei5π/62/2 0 0
)
54
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
e-iπ/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
e-iπ/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
0 0 eiπ/62/2 eiπ/62/2
0 0 eiπ/62/2 e-i5π/62/2
e-i2π/32/2 ei5π/62/2 0 0
ei5π/62/2 e-i2π/32/2 0 0
)
55
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
ei3π/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
ei3π/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
0 0 eiπ/62/2 e-i5π/62/2
0 0 e-i5π/62/2 e-i5π/62/2
e-i2π/32/2 e-iπ/62/2 0 0
e-iπ/62/2 e-i2π/32/2 0 0
)
56
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
e-i3π/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
e-i3π/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
0 0 e-iπ/32/2 e-iπ/32/2
0 0 ei2π/32/2 e-iπ/32/2
e-iπ/62/2 e-i2π/32/2 0 0
eiπ/32/2 ei5π/62/2 0 0
)
57
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
(
e-i3π/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
e-i3π/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
0 0 ei5π/62/2 eiπ/32/2
0 0 e-i2π/32/2 e-iπ/62/2
ei2π/32/2 ei2π/32/2 0 0
e-iπ/32/2 ei2π/32/2 0 0
)
58
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
e-iπ/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
e-iπ/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
0 0 eiπ/32/2 ei5π/62/2
0 0 ei5π/62/2 eiπ/32/2
e-i5π/62/2 eiπ/62/2 0 0
eiπ/62/2 eiπ/62/2 0 0
)
59
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
eiπ/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
eiπ/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
0 0 ei5π/62/2 e-i2π/32/2
0 0 eiπ/32/2 e-iπ/62/2
ei2π/32/2 e-iπ/32/2 0 0
ei2π/32/2 ei2π/32/2 0 0
)
60
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
ei3π/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
ei3π/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
0 0 eiπ/32/2 e-iπ/62/2
0 0 e-iπ/62/2 eiπ/32/2
e-i5π/62/2 e-i5π/62/2 0 0
e-i5π/62/2 eiπ/62/2 0 0
)
61
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2'110
1
-1
(
-i 0
0 -i
)
(
-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
)
(
e-i3π/4 0 0 0
0 ei3π/4 0 0
0 0 e-i3π/4 0
0 0 0 ei3π/4
)
62
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
2'1-10
1
-1
(
-i 0
0 -i
)
(
-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
)
(
ei3π/4 0 0 0
0 e-i3π/4 0 0
0 0 e-iπ/4 0
0 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
(
-i 0
0 -i
)
(
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 e-i3π/4 0 0
e-iπ/4 0 0 0
0 0 0 ei3π/4
0 0 eiπ/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
(
-i 0
0 -i
)
(
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 ei3π/4 0 0
eiπ/4 0 0 0
0 0 0 e-i3π/4
0 0 e-iπ/4 0
)
65
(
1 0 0
0 0 1
0 -1 0
)
(
2/2 i2/2
i2/2 2/2
)
4'-100
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
e-iπ/62/2 eiπ/32/2 0 0
e-i2π/32/2 ei5π/62/2 0 0
0 0 ei2π/32/2 e-iπ/32/2
0 0 ei2π/32/2 ei2π/32/2
)
66
(
-1 0 0
0 0 1
0 1 0
)
(
i2/2 2/2
-2/2 -i2/2
)
2'011
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
e-i2π/32/2 ei5π/62/2 0 0
ei5π/62/2 e-i2π/32/2 0 0
0 0 e-i5π/62/2 e-i5π/62/2
0 0 e-i5π/62/2 eiπ/62/2
)
67
(
-1 0 0
0 0 -1
0 -1 0
)
(
-i2/2 2/2
-2/2 i2/2
)
2'01-1
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
e-i2π/32/2 e-iπ/62/2 0 0
e-iπ/62/2 e-i2π/32/2 0 0
0 0 e-i5π/62/2 eiπ/62/2
0 0 eiπ/62/2 eiπ/62/2
)
68
(
1 0 0
0 0 -1
0 1 0
)
(
2/2 -i2/2
-i2/2 2/2
)
4'+100
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
ei5π/62/2 eiπ/32/2 0 0
e-i2π/32/2 e-iπ/62/2 0 0
0 0 e-iπ/32/2 e-iπ/32/2
0 0 ei2π/32/2 e-iπ/32/2
)
69
(
0 0 1
0 1 0
-1 0 0
)
(
2/2 -2/2
2/2 2/2
)
4'+010
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
e-iπ/32/2 e-iπ/32/2 0 0
ei2π/32/2 e-iπ/32/2 0 0
0 0 ei5π/62/2 eiπ/32/2
0 0 e-i2π/32/2 e-iπ/62/2
)
70
(
0 0 1
0 -1 0
1 0 0
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
2'101
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
eiπ/62/2 e-i5π/62/2 0 0
e-i5π/62/2 e-i5π/62/2 0 0
0 0 eiπ/32/2 ei5π/62/2
0 0 ei5π/62/2 eiπ/32/2
)
71
(
0 0 -1
0 1 0
1 0 0
)
(
2/2 2/2
-2/2 2/2
)
4'-010
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
ei2π/32/2 e-iπ/32/2 0 0
ei2π/32/2 ei2π/32/2 0 0
0 0 e-iπ/62/2 eiπ/32/2
0 0 e-i2π/32/2 ei5π/62/2
)
72
(
0 0 -1
0 -1 0
-1 0 0
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
2'-101
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
eiπ/62/2 eiπ/62/2 0 0
eiπ/62/2 e-i5π/62/2 0 0
0 0 eiπ/32/2 e-iπ/62/2
0 0 e-iπ/62/2 eiπ/32/2
)
73
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1'
1
1
(
0 -i
-i 0
)
(
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 0 eiπ/4 0
0 0 0 e-iπ/4
e-i3π/4 0 0 0
0 ei3π/4 0 0
)
74
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2'001
1
1
(
0 -i
-i 0
)
(
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 0 e-iπ/4 0
0 0 0 eiπ/4
e-iπ/4 0 0 0
0 eiπ/4 0 0
)
75
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2'010
1
1
(
0 -i
-i 0
)
(
-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
)
(
0 0 0 ei3π/4
0 0 eiπ/4 0
0 eiπ/4 0 0
ei3π/4 0 0 0
)
76
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2'100
1
1
(
0 -i
-i 0
)
(
-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
)
(
0 0 0 e-i3π/4
0 0 e-iπ/4 0
0 e-iπ/4 0 0
e-i3π/4 0 0 0
)
77
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
(
e-i3π/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
e-i3π/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
0 0 ei2π/32/2 e-iπ/32/2
0 0 ei2π/32/2 ei2π/32/2
ei5π/62/2 e-i2π/32/2 0 0
eiπ/32/2 e-iπ/62/2 0 0
)
78
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
ei3π/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
ei3π/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
0 0 e-i5π/62/2 e-i5π/62/2
0 0 e-i5π/62/2 eiπ/62/2
eiπ/32/2 e-iπ/62/2 0 0
e-iπ/62/2 eiπ/32/2 0 0
)
79
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
e-iπ/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
e-iπ/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
0 0 e-i5π/62/2 eiπ/62/2
0 0 eiπ/62/2 eiπ/62/2
eiπ/32/2 ei5π/62/2 0 0
ei5π/62/2 eiπ/32/2 0 0
)
80
(
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
(
0 eiπ/6
ei5π/6 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
eiπ/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
eiπ/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
0 0 ei2π/32/2 ei2π/32/2
0 0 e-iπ/32/2 ei2π/32/2
ei5π/62/2 eiπ/32/2 0 0
e-i2π/32/2 e-iπ/62/2 0 0
)
81
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
(
eiπ/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
eiπ/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
0 0 e-iπ/62/2 e-i2π/32/2
0 0 eiπ/32/2 ei5π/62/2
e-iπ/32/2 e-iπ/32/2 0 0
ei2π/32/2 e-iπ/32/2 0 0
)
82
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
ei3π/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
ei3π/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
0 0 e-i2π/32/2 e-iπ/62/2
0 0 e-iπ/62/2 e-i2π/32/2
eiπ/62/2 e-i5π/62/2 0 0
e-i5π/62/2 e-i5π/62/2 0 0
)
83
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
e-i3π/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
e-i3π/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
0 0 e-iπ/62/2 eiπ/32/2
0 0 e-i2π/32/2 ei5π/62/2
e-iπ/32/2 ei2π/32/2 0 0
e-iπ/32/2 e-iπ/32/2 0 0
)
84
(
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
(
0 ei5π/6
eiπ/6 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
e-iπ/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
e-iπ/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
0 0 e-i2π/32/2 ei5π/62/2
0 0 ei5π/62/2 e-i2π/32/2
eiπ/62/2 eiπ/62/2 0 0
eiπ/62/2 e-i5π/62/2 0 0
)
85
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2'110
1
-1
(
-i 0
0 -i
)
(
-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
)
(
eiπ/4 0 0 0
0 e-iπ/4 0 0
0 0 eiπ/4 0
0 0 0 e-iπ/4
)
86
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d2'1-10
1
-1
(
-i 0
0 -i
)
(
-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
)
(
e-iπ/4 0 0 0
0 eiπ/4 0 0
0 0 ei3π/4 0
0 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
(
-i 0
0 -i
)
(
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 eiπ/4 0 0
ei3π/4 0 0 0
0 0 0 e-iπ/4
0 0 e-i3π/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
(
-i 0
0 -i
)
(
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 e-iπ/4 0 0
e-i3π/4 0 0 0
0 0 0 eiπ/4
0 0 ei3π/4 0
)
89
(
1 0 0
0 0 1
0 -1 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
d4'-100
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
ei5π/62/2 e-i2π/32/2 0 0
eiπ/32/2 e-iπ/62/2 0 0
0 0 e-iπ/32/2 ei2π/32/2
0 0 e-iπ/32/2 e-iπ/32/2
)
90
(
-1 0 0
0 0 1
0 1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
d2'011
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
eiπ/32/2 e-iπ/62/2 0 0
e-iπ/62/2 eiπ/32/2 0 0
0 0 eiπ/62/2 eiπ/62/2
0 0 eiπ/62/2 e-i5π/62/2
)
91
(
-1 0 0
0 0 -1
0 -1 0
)
(
i2/2 -2/2
2/2 -i2/2
)
d2'01-1
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
eiπ/32/2 ei5π/62/2 0 0
ei5π/62/2 eiπ/32/2 0 0
0 0 eiπ/62/2 e-i5π/62/2
0 0 e-i5π/62/2 e-i5π/62/2
)
92
(
1 0 0
0 0 -1
0 1 0
)
(
-2/2 i2/2
i2/2 -2/2
)
d4'+100
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
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
)
(
e-iπ/62/2 e-i2π/32/2 0 0
eiπ/32/2 ei5π/62/2 0 0
0 0 ei2π/32/2 ei2π/32/2
0 0 e-iπ/32/2 ei2π/32/2
)
93
(
0 0 1
0 1 0
-1 0 0
)
(
-2/2 2/2
-2/2 -2/2
)
d4'+010
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
ei2π/32/2 ei2π/32/2 0 0
e-iπ/32/2 ei2π/32/2 0 0
0 0 e-iπ/62/2 e-i2π/32/2
0 0 eiπ/32/2 ei5π/62/2
)
94
(
0 0 1
0 -1 0
1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
d2'101
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
e-i5π/62/2 eiπ/62/2 0 0
eiπ/62/2 eiπ/62/2 0 0
0 0 e-i2π/32/2 e-iπ/62/2
0 0 e-iπ/62/2 e-i2π/32/2
)
95
(
0 0 -1
0 1 0
1 0 0
)
(
-2/2 -2/2
2/2 -2/2
)
d4'-010
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
e-iπ/32/2 ei2π/32/2 0 0
e-iπ/32/2 e-iπ/32/2 0 0
0 0 ei5π/62/2 e-i2π/32/2
0 0 eiπ/32/2 e-iπ/62/2
)
96
(
0 0 -1
0 -1 0
-1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
d2'-101
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
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
)
(
e-i5π/62/2 e-i5π/62/2 0 0
e-i5π/62/2 eiπ/62/2 0 0
0 0 e-i2π/32/2 ei5π/62/2
0 0 ei5π/62/2 e-i2π/32/2
)
k-Subgroupsmag
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