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Irreducible corepresentations of the Magnetic Point Group 6'/m' (N. 23.5.86)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM2+GM3+
2Eg1Eg
GM2+GM3+
2
-1
-1
2
-1
-1
2
-1
-1
2
-1
-1
GM2-GM3-
2Eu1Eu
GM2-GM3-
2
-1
-1
-2
1
1
2
-1
-1
-2
1
1
GM6+
Eg
GM4
1
-1
-1
1
-1
-1
-1
1
1
-1
1
1
GM4+GM5+
1Eg2Eg
GM5GM6
2
1
1
2
1
1
-2
-1
-1
-2
-1
-1
GM6-
Eu
GM7
1
-1
-1
-1
1
1
-1
1
1
1
-1
-1
GM4-GM5-
1Eu2Eu
GM8GM9
2
1
1
-2
-1
-1
-2
-1
-1
2
1
1
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: 3+001
C3: 3-001
C41
C53+001
C63-001
C7d1
C8d3+001
C9d3-001
C10d1
C11d3+001
C12d3-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM3+GM2-GM3-GM4GM5GM6GM7GM8GM9
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
1
(
1 0
0 1
)
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
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
-1
(
e-iπ/3 0
0 eiπ/3
)
-1
(
e-iπ/3 0
0 eiπ/3
)
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
-1
(
eiπ/3 0
0 e-iπ/3
)
-1
(
eiπ/3 0
0 e-iπ/3
)
4
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
1
(
1 0
0 1
)
-1
(
-1 0
0 -1
)
5
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
-1
(
e-iπ/3 0
0 eiπ/3
)
1
(
ei2π/3 0
0 e-i2π/3
)
6
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
-1
(
eiπ/3 0
0 e-iπ/3
)
1
(
e-i2π/3 0
0 ei2π/3
)
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
-1
(
-1 0
0 -1
)
-1
(
-1 0
0 -1
)
8
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
1
(
ei2π/3 0
0 e-i2π/3
)
1
(
ei2π/3 0
0 e-i2π/3
)
9
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
1
(
e-i2π/3 0
0 ei2π/3
)
1
(
e-i2π/3 0
0 ei2π/3
)
10
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
-1
(
-1 0
0 -1
)
1
(
1 0
0 1
)
11
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
1
(
ei2π/3 0
0 e-i2π/3
)
-1
(
e-iπ/3 0
0 eiπ/3
)
12
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
1
(
e-i2π/3 0
0 ei2π/3
)
-1
(
eiπ/3 0
0 e-iπ/3
)
13
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2'001
1
1
(
0 1
1 0
)
(
0 1
1 0
)
1
(
0 1
1 0
)
1
(
0 1
1 0
)
14
(
0 1 0
-1 1 0
0 0 1
)
(
(3-i)/2 0
0 (3+i)/2
)
6'-001
1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
-1
(
0 e-iπ/3
eiπ/3 0
)
-1
(
0 e-iπ/3
eiπ/3 0
)
15
(
1 -1 0
1 0 0
0 0 1
)
(
(3+i)/2 0
0 (3-i)/2
)
6'+001
1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
1
(
0 e-i2π/3
ei2π/3 0
)
1
(
0 e-i2π/3
ei2π/3 0
)
16
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m'001
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
1
(
0 1
1 0
)
-1
(
0 -1
-1 0
)
17
(
0 -1 0
1 -1 0
0 0 -1
)
(
(3-i)/2 0
0 (3+i)/2
)
6'-001
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
-1
(
0 e-iπ/3
eiπ/3 0
)
1
(
0 ei2π/3
e-i2π/3 0
)
18
(
-1 1 0
-1 0 0
0 0 -1
)
(
(3+i)/2 0
0 (3-i)/2
)
6'+001
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
1
(
0 e-i2π/3
ei2π/3 0
)
-1
(
0 eiπ/3
e-iπ/3 0
)
19
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2'001
1
1
(
0 1
1 0
)
(
0 1
1 0
)
-1
(
0 -1
-1 0
)
-1
(
0 -1
-1 0
)
20
(
0 1 0
-1 1 0
0 0 1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6'-001
1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
1
(
0 ei2π/3
e-i2π/3 0
)
1
(
0 ei2π/3
e-i2π/3 0
)
21
(
1 -1 0
1 0 0
0 0 1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6'+001
1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
-1
(
0 eiπ/3
e-iπ/3 0
)
-1
(
0 eiπ/3
e-iπ/3 0
)
22
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm'001
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
-1
(
0 -1
-1 0
)
1
(
0 1
1 0
)
23
(
0 -1 0
1 -1 0
0 0 -1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6'-001
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
1
(
0 ei2π/3
e-i2π/3 0
)
-1
(
0 e-iπ/3
eiπ/3 0
)
24
(
-1 1 0
-1 0 0
0 0 -1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6'+001
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
-1
(
0 eiπ/3
e-iπ/3 0
)
1
(
0 e-i2π/3
ei2π/3 0
)
k-Subgroupsmag
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