Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group -3'm' (N. 20.4.74)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
GM1
A1
GM1
1
1
1
1
1
1
GM2
A2
GM2
1
1
-1
1
1
-1
GM3
E
GM3
2
-1
0
2
-1
0
GM6GM5
2E1E
GM4GM5
2
-2
0
-2
2
0
GM4
E1
GM6
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
C4d1
C5d3+001d3-001
C6d2110d2120d2210

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
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
(
ei2π/3 0
0 e-i2π/3
)
(
-1 0
0 -1
)
(
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
(
e-i2π/3 0
0 ei2π/3
)
(
-1 0
0 -1
)
(
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
(
0 1
1 0
)
(
-i 0
0 i
)
(
0 -1
1 0
)
5
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
-i 0
0 i
)
(
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
(
0 ei2π/3
e-i2π/3 0
)
(
-i 0
0 i
)
(
0 e-iπ/3
e-i2π/3 0
)
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
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
)
d3+001
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
1 0
0 1
)
(
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
)
d3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
1 0
0 1
)
(
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
)
d21-10
1
-1
(
0 1
1 0
)
(
i 0
0 -i
)
(
0 1
-1 0
)
11
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
i 0
0 -i
)
(
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
)
d2210
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
i 0
0 -i
)
(
0 ei2π/3
eiπ/3 0
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1'
1
1
(
0 -i
-i 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
14
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3'+001
1
1
(
0 eiπ/6
ei5π/6 0
)
(
0 1
-1 0
)
(
0 e-iπ/3
e-i2π/3 0
)
15
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3'-001
1
1
(
0 ei5π/6
eiπ/6 0
)
(
0 1
-1 0
)
(
0 eiπ/3
ei2π/3 0
)
16
(
0 1 0
1 0 0
0 0 1
)
(
0 -(3-i)/2
(3+i)/2 0
)
m'1-10
1
-1
(
-i 0
0 -i
)
(
0 i
i 0
)
(
1 0
0 1
)
17
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m'120
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
0 i
i 0
)
(
e-i2π/3 0
0 ei2π/3
)
18
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m'210
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
0 i
i 0
)
(
ei2π/3 0
0 e-i2π/3
)
19
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1'
1
1
(
0 -i
-i 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
20
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3'+001
1
1
(
0 eiπ/6
ei5π/6 0
)
(
0 -1
1 0
)
(
0 ei2π/3
eiπ/3 0
)
21
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3'-001
1
1
(
0 ei5π/6
eiπ/6 0
)
(
0 -1
1 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
)
dm'1-10
1
-1
(
-i 0
0 -i
)
(
0 -i
-i 0
)
(
-1 0
0 -1
)
23
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm'120
1
-1
(
ei5π/6 0
0 eiπ/6
)
(
0 -i
-i 0
)
(
eiπ/3 0
0 e-iπ/3
)
24
(
-1 0 0
-1 1 0
0 0 1
)
(
0 -(3+i)/2
(3-i)/2 0
)
dm'210
1
-1
(
eiπ/6 0
0 ei5π/6
)
(
0 -i
-i 0
)
(
e-iπ/3 0
0 eiπ/3
)
k-Subgroupsmag
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