Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group -3m1' (N. 20.2.72)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM2+
A2g
GM2+
1
1
-1
1
1
-1
1
1
-1
1
1
-1
GM2-
A2u
GM2-
1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
GM3+
Eg
GM3+
2
-1
0
2
-1
0
2
-1
0
2
-1
0
GM3-
Eu
GM3-
2
-1
0
-2
1
0
2
-1
0
-2
1
0
GM4+GM5+
2Eg1Eg
GM4GM5
2
-2
0
2
-2
0
-2
2
0
-2
2
0
GM4-GM5-
2Eu1Eu
GM6GM7
2
-2
0
-2
2
0
-2
2
0
2
-2
0
GM6+
E1g
GM8
2
1
0
2
1
0
-2
-1
0
-2
-1
0
GM6-
E1u
GM9
2
1
0
-2
-1
0
-2
-1
0
2
1
0
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, 3-001
C3: 2110, 2120, 2210
C41
C53+0013-001
C6: m110, m120, m210
C7d1
C8d3+001d3-001
C9d2110d2120d2210
C10d1
C11d3+001d3-001
C12dm110dm120dm210

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM3-GM4GM5GM6GM7GM8GM9
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
)
(
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
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
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
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
4
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
21-10
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
0 -1
1 0
)
(
0 -1
1 0
)
5
(
-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 e-i2π/3
ei2π/3 0
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
0 eiπ/3
ei2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
6
(
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 ei2π/3
e-i2π/3 0
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
7
(
-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
)
(
-1 0
0 -1
)
8
(
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
)
(
e-iπ/3 0
0 eiπ/3
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
e-iπ/3 0
0 eiπ/3
)
(
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
)
3-001
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
10
(
0 1 0
1 0 0
0 0 1
)
(
0 -(3-i)/2
(3+i)/2 0
)
m1-10
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
0 -1
1 0
)
(
0 1
-1 0
)
11
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m120
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
12
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m210
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 ei2π/3
eiπ/3 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
)
(
-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
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/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
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
16
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d21-10
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
0 1
-1 0
)
(
0 1
-1 0
)
17
(
-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 e-i2π/3
ei2π/3 0
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
18
(
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 ei2π/3
e-i2π/3 0
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
0 ei2π/3
eiπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
19
(
-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
)
(
1 0
0 1
)
20
(
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
)
(
e-iπ/3 0
0 eiπ/3
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
21
(
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
)
(
eiπ/3 0
0 e-iπ/3
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
22
(
0 1 0
1 0 0
0 0 1
)
(
0 (3-i)/2
-(3+i)/2 0
)
dm1-10
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
0 1
-1 0
)
(
0 -1
1 0
)
23
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm120
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 eiπ/3
ei2π/3 0
)
24
(
-1 0 0
-1 1 0
0 0 1
)
(
0 -(3+i)/2
(3-i)/2 0
)
dm210
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
25
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1'
1
-1
1
-1
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
26
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3'+001
1
-1
1
-1
(
0 eiπ/6
ei5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 ei2π/3
eiπ/3 0
)
27
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3'-001
1
-1
1
-1
(
0 ei5π/6
eiπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
28
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2'1-10
1
-1
-1
1
(
-i 0
0 -i
)
(
i 0
0 i
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
1 0
0 1
)
(
-1 0
0 -1
)
29
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2'120
1
-1
-1
1
(
ei5π/6 0
0 eiπ/6
)
(
e-iπ/6 0
0 e-i5π/6
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
30
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2'210
1
-1
-1
1
(
eiπ/6 0
0 ei5π/6
)
(
e-i5π/6 0
0 e-iπ/6
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
31
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1'
1
1
1
1
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
32
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3'+001
1
1
1
1
(
0 eiπ/6
ei5π/6 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
33
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3'-001
1
1
1
1
(
0 ei5π/6
eiπ/6 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
34
(
0 1 0
1 0 0
0 0 1
)
(
0 -(3-i)/2
(3+i)/2 0
)
m'1-10
1
1
-1
-1
(
-i 0
0 -i
)
(
-i 0
0 -i
)
(
0 i
i 0
)
(
0 i
i 0
)
(
1 0
0 1
)
(
1 0
0 1
)
35
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m'120
1
1
-1
-1
(
ei5π/6 0
0 eiπ/6
)
(
ei5π/6 0
0 eiπ/6
)
(
0 i
i 0
)
(
0 i
i 0
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
36
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m'210
1
1
-1
-1
(
eiπ/6 0
0 ei5π/6
)
(
eiπ/6 0
0 ei5π/6
)
(
0 i
i 0
)
(
0 i
i 0
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
37
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1'
1
-1
1
-1
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
38
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3'+001
1
-1
1
-1
(
0 eiπ/6
ei5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
39
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3'-001
1
-1
1
-1
(
0 ei5π/6
eiπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 eiπ/3
ei2π/3 0
)
40
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2'1-10
1
-1
-1
1
(
-i 0
0 -i
)
(
i 0
0 i
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
-1 0
0 -1
)
(
1 0
0 1
)
41
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2'120
1
-1
-1
1
(
ei5π/6 0
0 eiπ/6
)
(
e-iπ/6 0
0 e-i5π/6
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
42
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2'210
1
-1
-1
1
(
eiπ/6 0
0 ei5π/6
)
(
e-i5π/6 0
0 e-iπ/6
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
43
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1'
1
1
1
1
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
44
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3'+001
1
1
1
1
(
0 eiπ/6
ei5π/6 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
45
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3'-001
1
1
1
1
(
0 ei5π/6
eiπ/6 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
46
(
0 1 0
1 0 0
0 0 1
)
(
0 (3-i)/2
-(3+i)/2 0
)
dm'1-10
1
1
-1
-1
(
-i 0
0 -i
)
(
-i 0
0 -i
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
47
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm'120
1
1
-1
-1
(
ei5π/6 0
0 eiπ/6
)
(
ei5π/6 0
0 eiπ/6
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
48
(
-1 0 0
-1 1 0
0 0 1
)
(
0 -(3+i)/2
(3-i)/2 0
)
dm'210
1
1
-1
-1
(
eiπ/6 0
0 ei5π/6
)
(
eiπ/6 0
0 ei5π/6
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
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