Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 321' (N. 18.2.66)


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
)
2'1-10
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
)
2'120
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
)
2'210
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
)
d2'1-10
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
)
d2'120
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
)
d2'210
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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