Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Double Point Group 43m (No. 31)

Table of characters

(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
GM5
T2
GM4
3
-1
0
1
-1
3
0
-1
GM4
T1
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
(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: m1-10, m110, m01-1, m011, m-101, m101dm1-10dm110dm01-1dm011dm-101dm101
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
C8d-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)
GM2(1)
GM3(1)
GM4(1)
GM5(1)
GM6(-1)
GM7(-1)
GM8(-1)
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 e-i5π/122/2
0 0 eiπ/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
(
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 eiπ/122/2
0 0 e-i5π/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
(
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 e-i11π/122/2
0 0 ei7π/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
(
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 ei7π/122/2
0 0 e-i11π/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 e-iπ/122/2
0 0 ei5π/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
(
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 e-i7π/122/2
0 0 ei11π/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
(
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 ei11π/122/2
0 0 e-i7π/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
(
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 ei5π/122/2
0 0 e-iπ/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
)
m110
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 -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
)
m110
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 -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
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/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
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/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
)
(
-i2/2 2/2
-2/2 i2/2
)
m011
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
)
(
-i2/2 2/2
-2/2 i2/2
)
(
i2/2 -2/2
2/2 -i2/2
)
(
0 0 e-i5π/122/2 e-i5π/122/2
0 0 eiπ/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
)
(
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
)
(
2/2 -i2/2
-i2/2 2/2
)
(
-2/2 i2/2
i2/2 -2/2
)
(
0 0 e-i11π/122/2 eiπ/122/2
0 0 e-i5π/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
)
(
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
)
(
2/2 i2/2
i2/2 2/2
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
0 0 eiπ/122/2 eiπ/122/2
0 0 e-i5π/122/2 ei7π/122/2
ei11π/122/2 ei5π/122/2 0 0
e-iπ/122/2 ei5π/122/2 0 0
)
20
(
1 0 0
0 0 -1
0 -1 0
)
(
i2/2 2/2
-2/2 -i2/2
)
m011
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
)
(
i2/2 2/2
-2/2 -i2/2
)
(
-i2/2 -2/2
2/2 i2/2
)
(
0 0 e-i5π/122/2 ei7π/122/2
0 0 e-i11π/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
)
21
(
0 0 1
0 1 0
1 0 0
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
m101
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
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
0 0 ei5π/122/2 e-iπ/122/2
0 0 ei5π/122/2 ei11π/122/2
ei7π/122/2 ei7π/122/2 0 0
e-i11π/122/2 eiπ/122/2 0 0
)
22
(
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
)
(
2/2 2/2
-2/2 2/2
)
(
-2/2 -2/2
2/2 -2/2
)
(
0 0 ei11π/122/2 e-i7π/122/2
0 0 ei11π/122/2 ei5π/122/2
eiπ/122/2 e-i11π/122/2 0 0
ei7π/122/2 ei7π/122/2 0 0
)
23
(
0 0 -1
0 1 0
-1 0 0
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
m101
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
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
0 0 ei5π/122/2 ei11π/122/2
0 0 e-i7π/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
)
(
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
)
(
2/2 -2/2
2/2 2/2
)
(
-2/2 2/2
-2/2 -2/2
)
(
0 0 e-iπ/122/2 e-i7π/122/2
0 0 ei11π/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 ei7π/122/2
0 0 e-i11π/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+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-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 e-i11π/122/2
0 0 ei7π/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+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-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 eiπ/122/2
0 0 e-i5π/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+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 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 e-i5π/122/2
0 0 eiπ/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 ei11π/122/2
0 0 e-i7π/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
)
d3-111
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
)
(
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 ei5π/122/2
0 0 e-iπ/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
)
d3-111
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 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 e-iπ/122/2
0 0 ei5π/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
)
d3-111
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
)
(
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 e-i7π/122/2
0 0 ei11π/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
)
dm110
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 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
)
dm110
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 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
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/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
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/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
)
(
i2/2 -2/2
2/2 -i2/2
)
dm011
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
)
(
i2/2 -2/2
2/2 -i2/2
)
(
-i2/2 2/2
-2/2 i2/2
)
(
0 0 ei7π/122/2 ei7π/122/2
0 0 e-i11π/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
)
(
-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
)
(
-2/2 i2/2
i2/2 -2/2
)
(
2/2 -i2/2
-i2/2 2/2
)
(
0 0 eiπ/122/2 e-i11π/122/2
0 0 ei7π/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
)
(
-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
)
(
-2/2 -i2/2
-i2/2 -2/2
)
(
2/2 i2/2
i2/2 2/2
)
(
0 0 e-i11π/122/2 e-i11π/122/2
0 0 ei7π/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
)
44
(
1 0 0
0 0 -1
0 -1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
dm011
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
)
(
-i2/2 -2/2
2/2 i2/2
)
(
i2/2 2/2
-2/2 -i2/2
)
(
0 0 ei7π/122/2 e-i5π/122/2
0 0 eiπ/122/2 eiπ/122/2
ei5π/122/2 ei11π/122/2 0 0
e-i7π/122/2 ei11π/122/2 0 0
)
45
(
0 0 1
0 1 0
1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
dm101
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
)
(
-i2/2 i2/2
i2/2 i2/2
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
(
0 0 e-i7π/122/2 ei11π/122/2
0 0 e-i7π/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
)
46
(
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
)
(
-2/2 -2/2
2/2 -2/2
)
(
2/2 2/2
-2/2 2/2
)
(
0 0 e-iπ/122/2 ei5π/122/2
0 0 e-iπ/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
)
47
(
0 0 -1
0 1 0
-1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
dm101
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
)
(
i2/2 i2/2
i2/2 -i2/2
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
(
0 0 e-i7π/122/2 e-iπ/122/2
0 0 ei5π/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
)
(
-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
)
(
-2/2 2/2
-2/2 -2/2
)
(
2/2 -2/2
2/2 2/2
)
(
0 0 ei11π/122/2 ei5π/122/2
0 0 e-iπ/122/2 ei5π/122/2
eiπ/122/2 eiπ/122/2 0 0
e-i5π/122/2 ei7π/122/2 0 0
)
k-Subgroupsmag
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