Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group -4'3m' (N. 31.3.117)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A
GM1
1
1
1
1
1
1
1
GM2
1E
GM2
1
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
GM3
2E
GM3
1
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
GM4
T
GM4
3
-1
0
0
3
0
0
GM5
E
GM5
2
0
1
1
-2
-1
-1
GM7
2F
GM6
2
0
-(1-i3)/2
-(1+i3)/2
-2
(1-i3)/2
(1+i3)/2
GM6
1F
GM7
2
0
-(1+i3)/2
-(1-i3)/2
-2
(1+i3)/2
(1-i3)/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: 2001, 2010, 2100d2001d2010d2100
C3: 3+111, 3+111, 3+111, 3+111
C4: 3-111, 3-111, 3-111, 3-111
C5d1
C6d3+111d3+111d3+111d3+111
C7d3-111d3-111d3-111d3-111

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 0 0
0 1 0
0 0 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 0 0
0 -1 0
0 0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
3
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2010
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
4
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2100
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
5
(
0 0 1
1 0 0
0 1 0
)
(
(1-i)/2 -(1+i)/2
(1-i)/2 (1+i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
(
e-iπ/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
ei5π/122/2 e-iπ/122/2
ei5π/122/2 ei11π/122/2
)
(
e-i11π/122/2 ei7π/122/2
e-i11π/122/2 e-i5π/122/2
)
6
(
0 0 1
-1 0 0
0 -1 0
)
(
(1+i)/2 -(1-i)/2
(1+i)/2 (1-i)/2
)
3+-11-1
1
ei2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
(
eiπ/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
ei11π/122/2 e-i7π/122/2
ei11π/122/2 ei5π/122/2
)
(
e-i5π/122/2 eiπ/122/2
e-i5π/122/2 e-i11π/122/2
)
7
(
0 0 -1
-1 0 0
0 1 0
)
(
(1+i)/2 (1-i)/2
-(1+i)/2 (1-i)/2
)
3+1-1-1
1
ei2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
(
eiπ/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
ei11π/122/2 ei5π/122/2
e-iπ/122/2 ei5π/122/2
)
(
e-i5π/122/2 e-i11π/122/2
ei7π/122/2 e-i11π/122/2
)
8
(
0 0 -1
1 0 0
0 -1 0
)
(
(1-i)/2 (1+i)/2
-(1-i)/2 (1+i)/2
)
3+-1-11
1
ei2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
(
e-iπ/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
ei5π/122/2 ei11π/122/2
e-i7π/122/2 ei11π/122/2
)
(
e-i11π/122/2 e-i5π/122/2
eiπ/122/2 e-i5π/122/2
)
9
(
0 1 0
0 0 1
1 0 0
)
(
(1+i)/2 (1+i)/2
-(1-i)/2 (1-i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
(
eiπ/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
e-i5π/122/2 e-i5π/122/2
eiπ/122/2 e-i11π/122/2
)
(
ei11π/122/2 ei11π/122/2
e-i7π/122/2 ei5π/122/2
)
10
(
0 -1 0
0 0 1
-1 0 0
)
(
(1-i)/2 -(1-i)/2
(1+i)/2 (1+i)/2
)
3-1-1-1
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
(
e-iπ/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
e-i11π/122/2 eiπ/122/2
e-i5π/122/2 e-i5π/122/2
)
(
ei5π/122/2 e-i7π/122/2
ei11π/122/2 ei11π/122/2
)
11
(
0 1 0
0 0 -1
-1 0 0
)
(
(1+i)/2 -(1+i)/2
(1-i)/2 (1-i)/2
)
3--1-11
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
(
eiπ/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
e-i5π/122/2 ei7π/122/2
e-i11π/122/2 e-i11π/122/2
)
(
ei11π/122/2 e-iπ/122/2
ei5π/122/2 ei5π/122/2
)
12
(
0 -1 0
0 0 -1
1 0 0
)
(
(1-i)/2 (1-i)/2
-(1+i)/2 (1+i)/2
)
3--11-1
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
(
e-iπ/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
e-i11π/122/2 e-i11π/122/2
ei7π/122/2 e-i5π/122/2
)
(
ei5π/122/2 ei5π/122/2
e-iπ/122/2 ei11π/122/2
)
13
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
14
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
(
1 0 0
0 -1 0
0 0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
15
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
17
(
0 0 1
1 0 0
0 1 0
)
(
-(1-i)/2 (1+i)/2
-(1-i)/2 -(1+i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
(
ei3π/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
e-i7π/122/2 ei11π/122/2
e-i7π/122/2 e-iπ/122/2
)
(
eiπ/122/2 e-i5π/122/2
eiπ/122/2 ei7π/122/2
)
18
(
0 0 1
-1 0 0
0 -1 0
)
(
-(1+i)/2 (1-i)/2
-(1+i)/2 -(1-i)/2
)
d3+-11-1
1
ei2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
(
e-i3π/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
e-iπ/122/2 ei5π/122/2
e-iπ/122/2 e-i7π/122/2
)
(
ei7π/122/2 e-i11π/122/2
ei7π/122/2 eiπ/122/2
)
19
(
0 0 -1
-1 0 0
0 1 0
)
(
-(1+i)/2 -(1-i)/2
(1+i)/2 -(1-i)/2
)
d3+1-1-1
1
ei2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
(
e-i3π/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
e-iπ/122/2 e-i7π/122/2
ei11π/122/2 e-i7π/122/2
)
(
ei7π/122/2 eiπ/122/2
e-i5π/122/2 eiπ/122/2
)
20
(
0 0 -1
1 0 0
0 -1 0
)
(
-(1-i)/2 -(1+i)/2
(1-i)/2 -(1+i)/2
)
d3+-1-11
1
ei2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
(
ei3π/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
e-i7π/122/2 e-iπ/122/2
ei5π/122/2 e-iπ/122/2
)
(
eiπ/122/2 ei7π/122/2
e-i11π/122/2 ei7π/122/2
)
21
(
0 1 0
0 0 1
1 0 0
)
(
-(1+i)/2 -(1+i)/2
(1-i)/2 -(1-i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
(
e-i3π/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
ei7π/122/2 ei7π/122/2
e-i11π/122/2 eiπ/122/2
)
(
e-iπ/122/2 e-iπ/122/2
ei5π/122/2 e-i7π/122/2
)
22
(
0 -1 0
0 0 1
-1 0 0
)
(
-(1-i)/2 (1-i)/2
-(1+i)/2 -(1+i)/2
)
d3-1-1-1
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
(
ei3π/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
eiπ/122/2 e-i11π/122/2
ei7π/122/2 ei7π/122/2
)
(
e-i7π/122/2 ei5π/122/2
e-iπ/122/2 e-iπ/122/2
)
23
(
0 1 0
0 0 -1
-1 0 0
)
(
-(1+i)/2 (1+i)/2
-(1-i)/2 -(1-i)/2
)
d3--1-11
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
(
e-i3π/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
ei7π/122/2 e-i5π/122/2
eiπ/122/2 eiπ/122/2
)
(
e-iπ/122/2 ei11π/122/2
e-i7π/122/2 e-i7π/122/2
)
24
(
0 -1 0
0 0 -1
1 0 0
)
(
-(1-i)/2 -(1-i)/2
(1+i)/2 -(1+i)/2
)
d3--11-1
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
(
ei3π/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
eiπ/122/2 eiπ/122/2
e-i5π/122/2 ei7π/122/2
)
(
e-i7π/122/2 e-i7π/122/2
ei11π/122/2 e-iπ/122/2
)
25
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m'1-10
1
1
1
(
-1 0 0
0 0 -1
0 -1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
ei3π/4 0
0 e-i3π/4
)
26
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m'110
1
1
1
(
-1 0 0
0 0 1
0 1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-i3π/4 0
0 ei3π/4
)
27
(
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 0
0 0 -1
0 1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 ei3π/4
eiπ/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
)
4'-001
1
1
1
(
1 0 0
0 0 1
0 -1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
29
(
1 0 0
0 0 1
0 1 0
)
(
-i2/2 2/2
-2/2 i2/2
)
m'01-1
1
e-i2π/3
ei2π/3
(
0 0 -1
0 -1 0
-1 0 0
)
(
1/2 i/2
i/2 1/2
)
(
e-i2π/32/2 e-iπ/62/2
e-iπ/62/2 e-i2π/32/2
)
(
ei2π/32/2 e-i5π/62/2
e-i5π/62/2 ei2π/32/2
)
30
(
-1 0 0
0 0 1
0 -1 0
)
(
2/2 -i2/2
-i2/2 2/2
)
4'+100
1
e-i2π/3
ei2π/3
(
0 0 1
0 1 0
-1 0 0
)
(
-i/2 -1/2
1/2 i/2
)
(
ei5π/62/2 eiπ/32/2
e-i2π/32/2 e-iπ/62/2
)
(
eiπ/62/2 e-iπ/32/2
ei2π/32/2 e-i5π/62/2
)
31
(
-1 0 0
0 0 -1
0 1 0
)
(
2/2 i2/2
i2/2 2/2
)
4'-100
1
e-i2π/3
ei2π/3
(
0 0 -1
0 1 0
1 0 0
)
(
i/2 -1/2
1/2 -i/2
)
(
e-iπ/62/2 eiπ/32/2
e-i2π/32/2 ei5π/62/2
)
(
e-i5π/62/2 e-iπ/32/2
ei2π/32/2 eiπ/62/2
)
32
(
1 0 0
0 0 -1
0 -1 0
)
(
i2/2 2/2
-2/2 -i2/2
)
m'011
1
e-i2π/3
ei2π/3
(
0 0 1
0 -1 0
1 0 0
)
(
1/2 -i/2
-i/2 1/2
)
(
e-i2π/32/2 ei5π/62/2
ei5π/62/2 e-i2π/32/2
)
(
ei2π/32/2 eiπ/62/2
eiπ/62/2 ei2π/32/2
)
33
(
0 0 1
0 1 0
1 0 0
)
(
i2/2 -i2/2
-i2/2 -i2/2
)
m'-101
1
ei2π/3
e-i2π/3
(
0 -1 0
-1 0 0
0 0 -1
)
(
-i/2 -i/2
-i/2 i/2
)
(
eiπ/62/2 eiπ/62/2
eiπ/62/2 e-i5π/62/2
)
(
ei5π/62/2 ei5π/62/2
ei5π/62/2 e-iπ/62/2
)
34
(
0 0 1
0 -1 0
-1 0 0
)
(
2/2 2/2
-2/2 2/2
)
4'-010
1
ei2π/3
e-i2π/3
(
0 1 0
-1 0 0
0 0 1
)
(
1/2 -1/2
1/2 1/2
)
(
ei2π/32/2 e-iπ/32/2
ei2π/32/2 ei2π/32/2
)
(
e-i2π/32/2 eiπ/32/2
e-i2π/32/2 e-i2π/32/2
)
35
(
0 0 -1
0 1 0
-1 0 0
)
(
-i2/2 -i2/2
-i2/2 i2/2
)
m'101
1
ei2π/3
e-i2π/3
(
0 1 0
1 0 0
0 0 -1
)
(
-i/2 i/2
i/2 i/2
)
(
eiπ/62/2 e-i5π/62/2
e-i5π/62/2 e-i5π/62/2
)
(
ei5π/62/2 e-iπ/62/2
e-iπ/62/2 e-iπ/62/2
)
36
(
0 0 -1
0 -1 0
1 0 0
)
(
2/2 -2/2
2/2 2/2
)
4'+010
1
ei2π/3
e-i2π/3
(
0 -1 0
1 0 0
0 0 1
)
(
-1/2 -1/2
1/2 -1/2
)
(
e-iπ/32/2 e-iπ/32/2
ei2π/32/2 e-iπ/32/2
)
(
eiπ/32/2 eiπ/32/2
e-i2π/32/2 eiπ/32/2
)
37
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm'1-10
1
1
1
(
-1 0 0
0 0 -1
0 -1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-iπ/4 0
0 eiπ/4
)
38
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm'110
1
1
1
(
-1 0 0
0 0 1
0 1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
eiπ/4 0
0 e-iπ/4
)
39
(
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 0
0 0 -1
0 1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
40
(
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 0
0 0 1
0 -1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 eiπ/4
ei3π/4 0
)
41
(
1 0 0
0 0 1
0 1 0
)
(
i2/2 -2/2
2/2 -i2/2
)
dm'01-1
1
e-i2π/3
ei2π/3
(
0 0 -1
0 -1 0
-1 0 0
)
(
-1/2 -i/2
-i/2 -1/2
)
(
eiπ/32/2 ei5π/62/2
ei5π/62/2 eiπ/32/2
)
(
e-iπ/32/2 eiπ/62/2
eiπ/62/2 e-iπ/32/2
)
42
(
-1 0 0
0 0 1
0 -1 0
)
(
-2/2 i2/2
i2/2 -2/2
)
d4'+100
1
e-i2π/3
ei2π/3
(
0 0 1
0 1 0
-1 0 0
)
(
i/2 1/2
-1/2 -i/2
)
(
e-iπ/62/2 e-i2π/32/2
eiπ/32/2 ei5π/62/2
)
(
e-i5π/62/2 ei2π/32/2
e-iπ/32/2 eiπ/62/2
)
43
(
-1 0 0
0 0 -1
0 1 0
)
(
-2/2 -i2/2
-i2/2 -2/2
)
d4'-100
1
e-i2π/3
ei2π/3
(
0 0 -1
0 1 0
1 0 0
)
(
-i/2 1/2
-1/2 i/2
)
(
ei5π/62/2 e-i2π/32/2
eiπ/32/2 e-iπ/62/2
)
(
eiπ/62/2 ei2π/32/2
e-iπ/32/2 e-i5π/62/2
)
44
(
1 0 0
0 0 -1
0 -1 0
)
(
-i2/2 -2/2
2/2 i2/2
)
dm'011
1
e-i2π/3
ei2π/3
(
0 0 1
0 -1 0
1 0 0
)
(
-1/2 i/2
i/2 -1/2
)
(
eiπ/32/2 e-iπ/62/2
e-iπ/62/2 eiπ/32/2
)
(
e-iπ/32/2 e-i5π/62/2
e-i5π/62/2 e-iπ/32/2
)
45
(
0 0 1
0 1 0
1 0 0
)
(
-i2/2 i2/2
i2/2 i2/2
)
dm'-101
1
ei2π/3
e-i2π/3
(
0 -1 0
-1 0 0
0 0 -1
)
(
i/2 i/2
i/2 -i/2
)
(
e-i5π/62/2 e-i5π/62/2
e-i5π/62/2 eiπ/62/2
)
(
e-iπ/62/2 e-iπ/62/2
e-iπ/62/2 ei5π/62/2
)
46
(
0 0 1
0 -1 0
-1 0 0
)
(
-2/2 -2/2
2/2 -2/2
)
d4'-010
1
ei2π/3
e-i2π/3
(
0 1 0
-1 0 0
0 0 1
)
(
-1/2 1/2
-1/2 -1/2
)
(
e-iπ/32/2 ei2π/32/2
e-iπ/32/2 e-iπ/32/2
)
(
eiπ/32/2 e-i2π/32/2
eiπ/32/2 eiπ/32/2
)
47
(
0 0 -1
0 1 0
-1 0 0
)
(
i2/2 i2/2
i2/2 -i2/2
)
dm'101
1
ei2π/3
e-i2π/3
(
0 1 0
1 0 0
0 0 -1
)
(
i/2 -i/2
-i/2 -i/2
)
(
e-i5π/62/2 eiπ/62/2
eiπ/62/2 eiπ/62/2
)
(
e-iπ/62/2 ei5π/62/2
ei5π/62/2 ei5π/62/2
)
48
(
0 0 -1
0 -1 0
1 0 0
)
(
-2/2 2/2
-2/2 -2/2
)
d4'+010
1
ei2π/3
e-i2π/3
(
0 -1 0
1 0 0
0 0 1
)
(
1/2 1/2
-1/2 1/2
)
(
ei2π/32/2 ei2π/32/2
e-iπ/32/2 ei2π/32/2
)
(
e-i2π/32/2 e-i2π/32/2
eiπ/32/2 e-i2π/32/2
)
k-Subgroupsmag
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