Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group -6m21' (N. 26.2.96)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
GM1
A1'
GM1
1
1
1
1
1
1
1
1
1
GM3
A1''
GM2
1
1
1
1
-1
-1
1
1
1
GM4
A2''
GM3
1
1
-1
-1
1
-1
1
1
-1
GM2
A2'
GM4
1
1
-1
-1
-1
1
1
1
-1
GM6
E'
GM5
2
-1
2
-1
0
0
2
-1
-1
GM5
E''
GM6
2
-1
-2
1
0
0
2
-1
1
GM9
E3
GM7
2
-2
0
0
0
0
-2
2
0
GM8
E2
GM8
2
1
0
-3
0
0
-2
-1
3
GM7
E1
GM9
2
1
0
3
0
0
-2
-1
-3
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: m001dm001
C46-0016+001
C5: m110, m100, m010dm110dm100dm010
C6: 2110, 2120, 2210d2110d2120d2210
C7d1
C8d3+001d3-001
C9d6-001d6+001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7GM8GM9
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
)
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
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/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
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
4
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
1
-1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
5
(
0 -1 0
1 -1 0
0 0 -1
)
(
(3-i)/2 0
0 (3+i)/2
)
6-001
1
1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
6
(
-1 1 0
-1 0 0
0 0 -1
)
(
(3+i)/2 0
0 (3-i)/2
)
6+001
1
1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
7
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
m110
1
-1
1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
-1 1 0
0 1 0
0 0 1
)
(
0 -1
1 0
)
m100
1
-1
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
9
(
1 0 0
1 -1 0
0 0 1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
m010
1
-1
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
10
(
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
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
11
(
-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 eiπ/3
e-iπ/3 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
12
(
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 e-iπ/3
eiπ/3 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 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
)
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
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/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
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
16
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
1
-1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
17
(
0 -1 0
1 -1 0
0 0 -1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6-001
1
1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
18
(
-1 1 0
-1 0 0
0 0 -1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6+001
1
1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
19
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
dm110
1
-1
1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
20
(
-1 1 0
0 1 0
0 0 1
)
(
0 1
-1 0
)
dm100
1
-1
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
21
(
1 0 0
1 -1 0
0 0 1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
dm010
1
-1
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
22
(
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
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
23
(
-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 eiπ/3
e-iπ/3 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
24
(
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 e-iπ/3
eiπ/3 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 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
)
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 eiπ/6
ei5π/6 0
)
(
0 -1
1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
(
0 eiπ/3
ei2π/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 ei5π/6
eiπ/6 0
)
(
0 -1
1 0
)
(
0 eiπ/3
ei2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
28
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m'001
1
1
-1
-1
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
29
(
0 -1 0
1 -1 0
0 0 -1
)
(
(3-i)/2 0
0 (3+i)/2
)
6'-001
1
1
-1
-1
(
0 eiπ/6
ei5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 i
i 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
30
(
-1 1 0
-1 0 0
0 0 -1
)
(
(3+i)/2 0
0 (3-i)/2
)
6'+001
1
1
-1
-1
(
0 ei5π/6
eiπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 -i
-i 0
)
(
0 ei5π/6
eiπ/6 0
)
(
0 eiπ/6
ei5π/6 0
)
31
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
m'110
1
-1
1
-1
(
-i 0
0 -i
)
(
-i 0
0 -i
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
32
(
-1 1 0
0 1 0
0 0 1
)
(
0 -1
1 0
)
m'100
1
-1
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
ei5π/6 0
0 eiπ/6
)
(
-1 0
0 -1
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
33
(
1 0 0
1 -1 0
0 0 1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
m'010
1
-1
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
eiπ/6 0
0 ei5π/6
)
(
1 0
0 1
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
34
(
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
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
35
(
-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
)
(
-i 0
0 i
)
(
ei5π/6 0
0 e-i5π/6
)
(
eiπ/6 0
0 e-iπ/6
)
36
(
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
)
(
-i 0
0 i
)
(
eiπ/6 0
0 e-iπ/6
)
(
ei5π/6 0
0 e-i5π/6
)
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
)
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 eiπ/6
ei5π/6 0
)
(
0 1
-1 0
)
(
0 ei2π/3
eiπ/3 0
)
(
0 e-i2π/3
e-iπ/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 ei5π/6
eiπ/6 0
)
(
0 1
-1 0
)
(
0 e-i2π/3
e-iπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
40
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm'001
1
1
-1
-1
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
41
(
0 -1 0
1 -1 0
0 0 -1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6'-001
1
1
-1
-1
(
0 eiπ/6
ei5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
(
0 -i
-i 0
)
(
0 eiπ/6
ei5π/6 0
)
(
0 ei5π/6
eiπ/6 0
)
42
(
-1 1 0
-1 0 0
0 0 -1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6'+001
1
1
-1
-1
(
0 ei5π/6
eiπ/6 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 i
i 0
)
(
0 e-iπ/6
e-i5π/6 0
)
(
0 e-i5π/6
e-iπ/6 0
)
43
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
dm'110
1
-1
1
-1
(
-i 0
0 -i
)
(
-i 0
0 -i
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
44
(
-1 1 0
0 1 0
0 0 1
)
(
0 1
-1 0
)
dm'100
1
-1
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
ei5π/6 0
0 eiπ/6
)
(
1 0
0 1
)
(
e-i2π/3 0
0 ei2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
45
(
1 0 0
1 -1 0
0 0 1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
dm'010
1
-1
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
eiπ/6 0
0 ei5π/6
)
(
-1 0
0 -1
)
(
e-iπ/3 0
0 eiπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
46
(
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
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
47
(
-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
)
(
i 0
0 -i
)
(
e-iπ/6 0
0 eiπ/6
)
(
e-i5π/6 0
0 ei5π/6
)
48
(
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
)
(
i 0
0 -i
)
(
e-i5π/6 0
0 ei5π/6
)
(
e-iπ/6 0
0 eiπ/6
)
k-Subgroupsmag
Bilbao Crystallographic Server
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