Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Double Point Group m3m (No. 32)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
GM2+
A2g
GM2+
1
1
1
-1
-1
1
1
-1
1
1
1
-1
-1
1
1
-1
GM2-
A2u
GM2-
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
GM3+
Eg
GM3+
2
2
-1
0
0
2
-1
0
2
2
-1
0
0
2
-1
0
GM3-
Eu
GM3-
2
2
-1
0
0
2
-1
0
-2
-2
1
0
0
-2
1
0
GM4+
T1g
GM4+
3
-1
0
-1
1
3
0
1
3
-1
0
-1
1
3
0
1
GM4-
T1u
GM4-
3
-1
0
-1
1
3
0
1
-3
1
0
1
-1
-3
0
-1
GM5+
T2g
GM5+
3
-1
0
1
-1
3
0
-1
3
-1
0
1
-1
3
0
-1
GM5-
T2u
GM5-
3
-1
0
1
-1
3
0
-1
-3
1
0
-1
1
-3
0
1
GM6+
E1g
GM6
2
0
1
0
2
-2
-1
-2
2
0
1
0
2
-2
-1
-2
GM7+
E2g
GM7
2
0
1
0
-2
-2
-1
2
2
0
1
0
-2
-2
-1
2
GM6-
E1u
GM8
2
0
1
0
2
-2
-1
-2
-2
0
-1
0
-2
2
1
2
GM7-
E2u
GM9
2
0
1
0
-2
-2
-1
2
-2
0
-1
0
2
2
1
-2
GM8+
Fg
GM10
4
0
-1
0
0
-4
1
0
4
0
-1
0
0
-4
1
0
GM8-
Fu
GM11
4
0
-1
0
0
-4
1
0
-4
0
1
0
0
4
-1
0
(1): Notation of the irreps according to Koster GF, Dimmok JO, Wheeler RG and Statz H, (1963) Properties of the thirty-two point groups, M.I.T. Press, Cambridge, Mass.
(2): Notation of the irreps according to Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to C. J. Bradley, A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001, 2010, 2100d2001d2010d2100
C3: 3+111, 3+-11-1, 3+1-1-1, 3+-1-11, 3-111, 3-1-1-1, 3--1-11, 3--11-1
C4: 2110, 21-10, 2011, 201-1, 2101, 2-101d2110d21-10d2011d201-1d2101d2-101
C5: 4-001, 4+001, 4-100, 4+100, 4+010, 4-010
C6d1
C7d3+111d3+-11-1d3+1-1-1d3+-1-11d3-111d3-1-1-1d3--1-11d3--11-1
C8d4-001d4+001d4-100d4+100d4+010d4-010
C9: -1
C10: m001, m010, m100dm001dm010dm100
C11: -3+111, -3+-11-1, -3+1-1-1, -3+-1-11, -3-111, -3-1-1-1, -3--1-11, -3--11-1
C12: m110, m1-10, m011, m01-1, m101, m-101dm110dm1-10dm011dm01-1dm101dm-101
C13: -4-001, -4+001, -4-100, -4+100, -4+010, -4-010
C14d-1
C15d-3+111d-3+-11-1d-3+1-1-1d-3+-1-11d-3-111d-3-1-1-1d-3--1-11d-3--11-1
C16d-4-001d-4+001d-4-100d-4+100d-4+010d-4-010

Matrices of the representations of the group

The number in parentheses after the label of the irrep indicates the "reality" of the irrep: (1) for real, (-1) for pseudoreal and (0) for complex representations.

N
Matrix presentation
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5+(1)
GM5-(1)
GM6(-1)
GM7(-1)
GM8(-1)
GM9(-1)
GM10(-1)
GM11(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
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 1
)
(
1 0
0 1
)
(
1 0 0 0
0 1 0 0
0 0 1 0
0 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
1
(
1 0
0 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
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 0
0 i 0 0
0 0 i 0
0 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
1
(
1 0
0 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
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 0 0
1 0 0 0
0 0 0 -i
0 0 -i 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
1
(
1 0
0 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
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 0 0
-i 0 0 0
0 0 0 -1
0 0 1 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
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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-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
)
(
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
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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-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
)
(
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
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
-1 0 0
0 0 1
0 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 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
)
(
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
)
2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
-1 0 0
0 0 -1
0 -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 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
)
(
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
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
1 0 0
0 0 1
0 -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
)
(
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
)
(
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
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
1 0 0
0 0 -1
0 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
)
(
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
)
(
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
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
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
)
(
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
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
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
)
(
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
)
2011
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
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
)
(
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
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
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
)
(
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
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
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
)
(
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
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
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
)
(
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
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
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
)
(
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
)
2101
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
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
)
(
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
)
1
1
-1
1
-1
(
1 0
0 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
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 0
0 1 0 0
0 0 1 0
0 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
)
m001
1
-1
1
-1
(
1 0
0 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
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 0
0 i 0 0
0 0 i 0
0 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
)
m010
1
-1
1
-1
(
1 0
0 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
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 0 0
1 0 0 0
0 0 0 -i
0 0 -i 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
)
m100
1
-1
1
-1
(
1 0
0 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
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 0 0
-i 0 0 0
0 0 0 -1
0 0 1 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
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 -1 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
)
(
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
)
(
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
)
(
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
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 1 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
)
(
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
)
(
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
)
(
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
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 1 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
)
(
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
)
(
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
)
(
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
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 -1 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
)
(
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
)
(
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
)
(
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
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 1
1 0 0
)
(
0 -1 0
0 0 -1
-1 0 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
)
(
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
)
(
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
)
(
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
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 1 0
0 0 1
-1 0 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
)
(
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
)
(
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
)
(
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
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 1 0
0 0 -1
1 0 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
)
(
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
)
(
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
)
(
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
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 -1 0
0 0 1
1 0 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
)
(
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
)
(
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
)
(
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
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 -1
0 -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 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
)
(
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
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 1
0 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 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
)
(
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
)
4-001
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 -1
0 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
)
(
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
)
(
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
)
4+001
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 1
0 -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
)
(
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
)
(
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
)
4-100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
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
)
(
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
)
m011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
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
)
(
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
)
m011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
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
)
(
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
)
4+100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
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
)
(
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
)
4+010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 1
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
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
)
(
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
)
m101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 -1
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
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
)
(
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
)
4-010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 1
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
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
)
(
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
)
m101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
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
)
(
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
)
d1
1
1
1
1
(
1 0
0 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
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 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
)
(
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
)
50
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
(
1 0
0 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
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 0
0 -i 0 0
0 0 -i 0
0 0 0 i
)
(
i 0 0 0
0 -i 0 0
0 0 -i 0
0 0 0 i
)
51
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
1
1
(
1 0
0 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
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 0 0
-1 0 0 0
0 0 0 i
0 0 i 0
)
(
0 1 0 0
-1 0 0 0
0 0 0 i
0 0 i 0
)
52
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
1
1
(
1 0
0 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
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 0 0
i 0 0 0
0 0 0 1
0 0 -1 0
)
(
0 i 0 0
i 0 0 0
0 0 0 1
0 0 -1 0
)
53
(
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
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
54
(
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
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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-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
)
(
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
)
55
(
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
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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-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
)
(
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
)
56
(
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
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
57
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
58
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
59
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
60
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
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
)
(
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
)
(
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
)
(
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
)
61
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
-1 0 0
0 0 1
0 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 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
)
(
0 0 1 0
0 0 0 1
-1 0 0 0
0 -1 0 0
)
62
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
-1 0 0
0 0 -1
0 -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 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
)
(
0 0 -i 0
0 0 0 i
-i 0 0 0
0 i 0 0
)
63
(
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
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
1 0 0
0 0 1
0 -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
)
(
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
)
(
0 0 0 i
0 0 i 0
0 -1 0 0
1 0 0 0
)
64
(
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
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
1 0 0
0 0 -1
0 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
)
(
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
)
(
0 0 0 1
0 0 -1 0
0 -i 0 0
-i 0 0 0
)
65
(
1 0 0
0 0 1
0 -1 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
d4-100
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
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
)
(
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
)
66
(
-1 0 0
0 0 1
0 1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
d2011
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
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
)
(
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
)
67
(
-1 0 0
0 0 -1
0 -1 0
)
(
i2/2 -2/2
2/2 -i2/2
)
d2011
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
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
)
(
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
)
68
(
1 0 0
0 0 -1
0 1 0
)
(
-2/2 i2/2
i2/2 -2/2
)
d4+100
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
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
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
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
)
(
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
)
69
(
0 0 1
0 1 0
-1 0 0
)
(
-2/2 2/2
-2/2 -2/2
)
d4+010
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
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
)
(
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
)
70
(
0 0 1
0 -1 0
1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
d2101
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
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
)
(
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
)
71
(
0 0 -1
0 1 0
1 0 0
)
(
-2/2 -2/2
2/2 -2/2
)
d4-010
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
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
)
(
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
)
72
(
0 0 -1
0 -1 0
-1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
d2101
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
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
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
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
)
(
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
)
73
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0 0
0 1 0
0 0 1
)
(
-1 0 0
0 -1 0
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 1
)
(
1 0
0 1
)
(
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
)
(
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
)
74
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
(
1 0
0 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
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 0
0 -i 0 0
0 0 -i 0
0 0 0 i
)
(
-i 0 0 0
0 i 0 0
0 0 i 0
0 0 0 -i
)
75
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm010
1
-1
1
-1
(
1 0
0 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
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 0 0
-1 0 0 0
0 0 0 i
0 0 i 0
)
(
0 -1 0 0
1 0 0 0
0 0 0 -i
0 0 -i 0
)
76
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm100
1
-1
1
-1
(
1 0
0 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
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 0 0
i 0 0 0
0 0 0 1
0 0 -1 0
)
(
0 -i 0 0
-i 0 0 0
0 0 0 -1
0 0 1 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
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 -1 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
)
(
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
)
(
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
)
(
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
)
78
(
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
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 1 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
)
(
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
)
(
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
)
(
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
)
79
(
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
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 1 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
)
(
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
)
(
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
)
(
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
)
80
(
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
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 -1 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
)
(
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
)
(
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
)
(
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
)
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
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 1
1 0 0
)
(
0 -1 0
0 0 -1
-1 0 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
)
(
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
)
(
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
)
(
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
)
82
(
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
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 1 0
0 0 1
-1 0 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
)
(
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
)
(
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
)
(
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
)
83
(
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
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 1 0
0 0 -1
1 0 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
)
(
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
)
(
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
)
(
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
)
84
(
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
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 -1 0
0 0 1
1 0 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
)
(
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
)
(
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
)
(
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
)
85
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 -1
0 -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 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
)
(
0 0 -1 0
0 0 0 -1
1 0 0 0
0 1 0 0
)
86
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 1
0 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 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
)
(
0 0 i 0
0 0 0 -i
i 0 0 0
0 -i 0 0
)
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
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 -1
0 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
)
(
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
)
(
0 0 0 -i
0 0 -i 0
0 1 0 0
-1 0 0 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
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 1
0 -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
)
(
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
)
(
0 0 0 -1
0 0 1 0
0 i 0 0
i 0 0 0
)
89
(
-1 0 0
0 0 -1
0 1 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
d4-100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
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
)
(
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
)
90
(
1 0 0
0 0 -1
0 -1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
dm011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
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
)
(
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
)
91
(
1 0 0
0 0 1
0 1 0
)
(
i2/2 -2/2
2/2 -i2/2
)
dm011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
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
)
(
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
)
92
(
-1 0 0
0 0 1
0 -1 0
)
(
-2/2 i2/2
i2/2 -2/2
)
d4+100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
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
)
(
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
)
93
(
0 0 -1
0 -1 0
1 0 0
)
(
-2/2 2/2
-2/2 -2/2
)
d4+010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 1
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
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
)
(
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
)
94
(
0 0 -1
0 1 0
-1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
dm101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 -1
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
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
)
(
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
)
95
(
0 0 1
0 -1 0
-1 0 0
)
(
-2/2 -2/2
2/2 -2/2
)
d4-010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 1
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
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
)
(
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
)
96
(
0 0 1
0 1 0
1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
dm101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
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
)
(
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
)
k-Subgroupsmag
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