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Irreducible representations of the Double Point Group 622 (No. 24)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
GM1
A1
GM1
1
1
1
1
1
1
1
1
1
GM2
A2
GM2
1
1
1
1
-1
-1
1
1
1
GM4
B2
GM3
1
1
-1
-1
1
-1
1
1
-1
GM3
B1
GM4
1
1
-1
-1
-1
1
1
1
-1
GM6
E2
GM5
2
-1
2
-1
0
0
2
-1
-1
GM5
E1
GM6
2
-1
-2
1
0
0
2
-1
1
GM9
E3
GM7
2
-2
0
0
0
0
-2
2
0
GM8
E2
GM8
2
1
0
-3
0
0
-2
-1
3
GM7
E1
GM9
2
1
0
3
0
0
-2
-1
-3
(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Notation of the irreps according to 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): Notation of the irreps according to 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 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

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)
GM9(-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 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
(
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
)
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
1
1
(
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
)
4
(
-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
)
(
-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
(
ei2π/3 0
0 e-i2π/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
)
6
(
1 -1 0
1 0 0
0 0 1
)
(
(3+i)/2 0
0 (3-i)/2
)
6+001
1
1
-1
-1
(
e-i2π/3 0
0 ei2π/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
)
7
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
2110
1
-1
1
-1
(
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
(
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
)
9
(
-1 0 0
-1 1 0
0 0 -1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
2010
1
-1
1
-1
(
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
)
10
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 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
(
0 e-i2π/3
ei2π/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
)
12
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2210
1
-1
-1
1
(
0 ei2π/3
e-i2π/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
)
13
(
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 -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
)
d3+001
1
1
1
1
(
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
)
15
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
1
1
(
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
)
16
(
-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
)
(
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
)
d6-001
1
1
-1
-1
(
ei2π/3 0
0 e-i2π/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
)
18
(
1 -1 0
1 0 0
0 0 1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6+001
1
1
-1
-1
(
e-i2π/3 0
0 ei2π/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
)
19
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
d2110
1
-1
1
-1
(
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
)
d2100
1
-1
1
-1
(
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
)
21
(
-1 0 0
-1 1 0
0 0 -1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
d2010
1
-1
1
-1
(
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
)
22
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 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
)
d2120
1
-1
-1
1
(
0 e-i2π/3
ei2π/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
)
24
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2210
1
-1
-1
1
(
0 ei2π/3
e-i2π/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
)
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