Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Double Point Group 6/mmm (No. 27)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
GM1+
A1g
GM1+
1
1
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
-1
-1
GM2+
A2g
GM2+
1
1
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
-1
-1
GM4+
B2g
GM3+
1
1
-1
-1
1
-1
1
1
-1
1
1
-1
-1
1
-1
1
1
-1
GM4-
B2u
GM3-
1
1
-1
-1
1
-1
1
1
-1
-1
-1
1
1
-1
1
-1
-1
1
GM3+
B1g
GM4+
1
1
-1
-1
-1
1
1
1
-1
1
1
-1
-1
-1
1
1
1
-1
GM3-
B1u
GM4-
1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
GM6+
E2g
GM5+
2
-1
2
-1
0
0
2
-1
-1
2
-1
2
-1
0
0
2
-1
-1
GM6-
E2u
GM5-
2
-1
2
-1
0
0
2
-1
-1
-2
1
-2
1
0
0
-2
1
1
GM5+
E1g
GM6+
2
-1
-2
1
0
0
2
-1
1
2
-1
-2
1
0
0
2
-1
1
GM5-
E1u
GM6-
2
-1
-2
1
0
0
2
-1
1
-2
1
2
-1
0
0
-2
1
-1
GM9+
E3g
GM7
2
-2
0
0
0
0
-2
2
0
2
-2
0
0
0
0
-2
2
0
GM8+
E2g
GM8
2
1
0
-3
0
0
-2
-1
3
2
1
0
-3
0
0
-2
-1
3
GM7+
E1g
GM9
2
1
0
3
0
0
-2
-1
-3
2
1
0
3
0
0
-2
-1
-3
GM9-
E3u
GM10
2
-2
0
0
0
0
-2
2
0
-2
2
0
0
0
0
2
-2
0
GM8-
E2u
GM11
2
1
0
-3
0
0
-2
-1
3
-2
-1
0
3
0
0
2
1
-3
GM7-
E1u
GM12
2
1
0
3
0
0
-2
-1
-3
-2
-1
0
-3
0
0
2
1
3
(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: 3+001, 3-001
C3: 2001d2001
C4: 6-001, 6+001
C5: 2110, 2100, 2010d2110d2100d2010
C6: 21-10, 2120, 2210d21-10d2120d2210
C7d1
C8d3+001d3-001
C9d6-001d6+001
C10: -1
C11: -3+001, -3-001
C12: m001dm001
C13: -6-001, -6+001
C14: m110, m100, m010dm110dm100dm010
C15: m1-10, m120, m210dm1-10dm120dm210
C16d-1
C17d-3+001d-3-001
C18d-6-001d-6+001

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)
GM6-(1)
GM7(-1)
GM8(-1)
GM9(-1)
GM10(-1)
GM11(-1)
GM12(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
2
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
1
1
1
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
1
1
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
4
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2001
1
1
1
1
-1
-1
-1
-1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
5
(
0 1 0
-1 1 0
0 0 1
)
(
(3-i)/2 0
0 (3+i)/2
)
6-001
1
1
1
1
-1
-1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
6
(
1 -1 0
1 0 0
0 0 1
)
(
(3+i)/2 0
0 (3-i)/2
)
6+001
1
1
1
1
-1
-1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
7
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
2110
1
1
-1
-1
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
1 -1 0
0 -1 0
0 0 -1
)
(
0 -1
1 0
)
2100
1
1
-1
-1
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
9
(
-1 0 0
-1 1 0
0 0 -1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
2010
1
1
-1
-1
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
10
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2110
1
1
-1
-1
-1
-1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
11
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2120
1
1
-1
-1
-1
-1
1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
12
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2210
1
1
-1
-1
-1
-1
1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
14
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
-1
1
-1
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
15
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
-1
1
-1
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
16
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
-1
1
-1
1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
17
(
0 -1 0
1 -1 0
0 0 -1
)
(
(3-i)/2 0
0 (3+i)/2
)
6-001
1
-1
1
-1
-1
1
-1
1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
18
(
-1 1 0
-1 0 0
0 0 -1
)
(
(3+i)/2 0
0 (3-i)/2
)
6+001
1
-1
1
-1
-1
1
-1
1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
19
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
m110
1
-1
-1
1
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
20
(
-1 1 0
0 1 0
0 0 1
)
(
0 -1
1 0
)
m100
1
-1
-1
1
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
21
(
1 0 0
1 -1 0
0 0 1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
m010
1
-1
-1
1
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
22
(
0 1 0
1 0 0
0 0 1
)
(
0 -(3-i)/2
(3+i)/2 0
)
m110
1
-1
-1
1
-1
1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
23
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m120
1
-1
-1
1
-1
1
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
24
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m210
1
-1
-1
1
-1
1
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 0
)
25
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
26
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
1
1
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
27
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
1
1
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
28
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
-1
-1
-1
-1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
29
(
0 1 0
-1 1 0
0 0 1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6-001
1
1
1
1
-1
-1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
30
(
1 -1 0
1 0 0
0 0 1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6+001
1
1
1
1
-1
-1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
31
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
d2110
1
1
-1
-1
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
32
(
1 -1 0
0 -1 0
0 0 -1
)
(
0 1
-1 0
)
d2100
1
1
-1
-1
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
33
(
-1 0 0
-1 1 0
0 0 -1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
d2010
1
1
-1
-1
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
34
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2110
1
1
-1
-1
-1
-1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
35
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2120
1
1
-1
-1
-1
-1
1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
36
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2210
1
1
-1
-1
-1
-1
1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 0
)
37
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
38
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
-1
1
-1
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
39
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
-1
1
-1
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
40
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
-1
1
-1
1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
41
(
0 -1 0
1 -1 0
0 0 -1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6-001
1
-1
1
-1
-1
1
-1
1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
42
(
-1 1 0
-1 0 0
0 0 -1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6+001
1
-1
1
-1
-1
1
-1
1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
43
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
dm110
1
-1
-1
1
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
44
(
-1 1 0
0 1 0
0 0 1
)
(
0 1
-1 0
)
dm100
1
-1
-1
1
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
45
(
1 0 0
1 -1 0
0 0 1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
dm010
1
-1
-1
1
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
46
(
0 1 0
1 0 0
0 0 1
)
(
0 (3-i)/2
-(3+i)/2 0
)
dm110
1
-1
-1
1
-1
1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
47
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm120
1
-1
-1
1
-1
1
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
48
(
-1 0 0
-1 1 0
0 0 1
)
(
0 -(3+i)/2
(3-i)/2 0
)
dm210
1
-1
-1
1
-1
1
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
k-Subgroupsmag
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