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Irreducible corepresentations of the Magnetic Point Group -4m21' (N. 14.2.49)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A1
GM1
1
1
1
1
1
1
1
GM3
B1
GM2
1
1
-1
1
-1
1
-1
GM4
B2
GM3
1
1
1
-1
-1
1
1
GM2
A2
GM4
1
1
-1
-1
1
1
-1
GM5
E
GM5
2
-2
0
0
0
2
0
GM7
E2
GM6
2
0
-2
0
0
-2
2
GM6
E1
GM7
2
0
2
0
0
-2
-2
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: 2001d2001
C34+0014-001
C4: m010, m100dm010dm100
C5: 2110, 2110d2110d2110
C6d1
C7d4+001d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7
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
)
2
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2001
1
1
1
1
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
3
(
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
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
4
(
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
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m010
1
1
-1
-1
(
0 1
1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
6
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
7
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2110
1
-1
-1
1
(
1 0
0 -1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
21-10
1
-1
-1
1
(
-1 0
0 1
)
(
0 i
i 0
)
(
0 i
i 0
)
9
(
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
)
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
11
(
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
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
12
(
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
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
13
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm010
1
1
-1
-1
(
0 1
1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
14
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
15
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2110
1
-1
-1
1
(
1 0
0 -1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d21-10
1
-1
-1
1
(
-1 0
0 1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
17
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1'
1
1
1
1
(
1 0
0 1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
18
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2'001
1
1
1
1
(
-1 0
0 -1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
19
(
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 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
20
(
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 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
21
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m'010
1
1
-1
-1
(
0 1
1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
22
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m'100
1
1
-1
-1
(
0 -1
-1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
23
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2'110
1
-1
-1
1
(
1 0
0 -1
)
(
1 0
0 1
)
(
1 0
0 1
)
24
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
2'1-10
1
-1
-1
1
(
-1 0
0 1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
25
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1'
1
1
1
1
(
1 0
0 1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
26
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2'001
1
1
1
1
(
-1 0
0 -1
)
(
0 i
i 0
)
(
0 i
i 0
)
27
(
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 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
28
(
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 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
29
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm'010
1
1
-1
-1
(
0 1
1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
30
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm'100
1
1
-1
-1
(
0 -1
-1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
31
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2'110
1
-1
-1
1
(
1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
32
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d2'1-10
1
-1
-1
1
(
-1 0
0 1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
k-Subgroupsmag
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