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Irreducible corepresentations of the Magnetic Point Group 4221' (N. 12.2.41)


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
GM2
A2
GM3
1
1
1
-1
-1
1
1
GM4
B2
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
C3: 4+001, 4-001
C4: 2010, 2100d2010d2100
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
)
2010
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
)
2100
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
)
d2010
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
)
d2100
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
)
2'010
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
)
2'100
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
)
d2'010
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
)
d2'100
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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