Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Projective Magnetic Point Group 62π/31'


Table of characters of the unitary symmetry operations


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1E1
1
e2iπ/3
e-2iπ/3
1
e2iπ/3
e-2iπ/3
1
e2iπ/3
e-2iπ/3
1
e2iπ/3
e-2iπ/3
1E2
1
e-iπ/3
e-2iπ/3
-1
e2iπ/3
eiπ/3
1
e-iπ/3
e-2iπ/3
-1
e2iπ/3
eiπ/3
2E1A
2
e-iπ/3
eiπ/3
2
e-iπ/3
eiπ/3
2
e-iπ/3
eiπ/3
2
e-iπ/3
eiπ/3
2E2B
2
e2iπ/3
eiπ/3
-2
e-iπ/3
e-2iπ/3
2
e2iπ/3
eiπ/3
-2
e-iπ/3
e-2iπ/3
1E22E3
2
0
2eiπ/3
0
2e-iπ/3
0
-2
0
2e-2iπ/3
0
2e2iπ/3
0
1E12E2
2
3e2iπ/3
e-2iπ/3
0
e2iπ/3
3e-2iπ/3
-2
3e-iπ/3
eiπ/3
0
e-iπ/3
3eiπ/3
2E11E3
2
3e-iπ/3
e-2iπ/3
0
e2iπ/3
3eiπ/3
-2
3e2iπ/3
eiπ/3
0
e-iπ/3
3e-2iπ/3

Multiplication table of the symmetry operations


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1
1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
6+
6+
3+
d2
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
6+'
3+'
d2'
3-'
6-'
1'
d6+'
d3+'
2'
d3-'
d6-'
d1'
3+
3+
d2
d3-
6-
1
6+
d3+
2
3-
d6-
d1
d6+
3+'
d2'
d3-'
6-'
1'
6+'
d3+'
2'
3-'
d6-'
d1'
d6+'
2
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1'
6+'
3+'
3-
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
3-'
6-'
1'
d6+'
d3+'
2'
d3-'
d6-'
d1'
6+'
3+'
d2'
6-
6-
1
6+
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-
6-'
1'
6+'
d3+'
2'
3-'
d6-'
d1'
d6+'
3+'
d2'
d3-'
d1
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
2
3-
6-
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1'
6+'
3+'
2'
3-'
6-'
d6+
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
3-
6-
1
d6+'
d3+'
2'
d3-'
d6-'
d1'
6+'
3+'
d2'
3-'
6-'
1'
d3+
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
d3+'
2'
3-'
d6-'
d1'
d6+'
3+'
d2'
d3-'
6-'
1'
6+'
d2
d2
d3-
d6-
1
6+
3+
2
3-
6-
d1
d6+
d3+
d2'
d3-'
d6-'
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
d3-
d3-
d6-
d1
6+
3+
d2
3-
6-
1
d6+
d3+
2
d3-'
d6-'
d1'
6+'
3+'
d2'
3-'
6-'
1'
d6+'
d3+'
2'
d6-
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
d3+
2
3-
d6-'
d1'
d6+'
3+'
d2'
d3-'
6-'
1'
6+'
d3+'
2'
3-'
1'
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
2
3-
6-
6+'
6+'
3+'
d2'
3-'
6-'
1'
d6+'
d3+'
2'
d3-'
d6-'
d1'
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
3-
6-
1
3+'
3+'
d2'
d3-'
6-'
1'
6+'
d3+'
2'
3-'
d6-'
d1'
d6+'
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
2'
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1'
6+'
3+'
d2
d3-
d6-
1
6+
3+
2
3-
6-
d1
d6+
d3+
3-'
3-'
6-'
1'
d6+'
d3+'
2'
d3-'
d6-'
d1'
6+'
3+'
d2'
d3-
d6-
d1
6+
3+
d2
3-
6-
1
d6+
d3+
2
6-'
6-'
1'
6+'
d3+'
2'
3-'
d6-'
d1'
d6+'
3+'
d2'
d3-'
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
d3+
2
3-
d1'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1'
6+'
3+'
2'
3-'
6-'
1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
d6+'
d6+'
d3+'
2'
d3-'
d6-'
d1'
6+'
3+'
d2'
3-'
6-'
1'
6+
3+
d2
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
d3+'
d3+'
2'
3-'
d6-'
d1'
d6+'
3+'
d2'
d3-'
6-'
1'
6+'
3+
d2
d3-
6-
1
6+
d3+
2
3-
d6-
d1
d6+
d2'
d2'
d3-'
d6-'
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
d3-'
d3-'
d6-'
d1'
6+'
3+'
d2'
3-'
6-'
1'
d6+'
d3+'
2'
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
d6-'
d6-'
d1'
d6+'
3+'
d2'
d3-'
6-'
1'
6+'
d3+'
2'
3-'
6-
1
6+
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-

Table of projective phases in group multiplication


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1'
6+'
3+'
2'
3-'
6-'
d1'
d6+'
d3+'
d2'
d3-'
d6-'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
6+
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
3+
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3-
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
6-
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
d1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d6+
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
d3+
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
d2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d3-
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
d6-
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
1'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
6+'
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
3+'
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
2'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
3-'
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
6-'
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
d1'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d6+'
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
d3+'
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
d2'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d3-'
1
1
1
1
1
1
1
1
1
1
1
1
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
d6-'
1
1
1
1
1
1
1
1
1
1
1
1
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol1E11E22E1A2E2B1E22E31E12E22E11E3
1
(
1 0
0 1
)
(
1 0
0 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
(
1 -1
1 0
)
(
eiπ/6 0
0 e-iπ/6
)
6+
e2iπ/3
e-iπ/3
(
1 0
0 e-2iπ/3
)
(
-1 0
0 eiπ/3
)
(
eiπ/6 0
0 e-5iπ/6
)
(
i 0
0 e5iπ/6
)
(
-i 0
0 e-iπ/6
)
3
(
0 -1
1 -1
)
(
eiπ/3 0
0 e-iπ/3
)
3+
e-2iπ/3
e-2iπ/3
(
1 0
0 e2iπ/3
)
(
1 0
0 e2iπ/3
)
(
eiπ/3 0
0 eiπ/3
)
(
-1 0
0 e-iπ/3
)
(
-1 0
0 e-iπ/3
)
4
(
-1 0
0 -1
)
(
-i 0
0 i
)
2
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
-i 0
0 i
)
5
(
-1 1
-1 0
)
(
e-iπ/3 0
0 eiπ/3
)
3-
e2iπ/3
e2iπ/3
(
1 0
0 e-2iπ/3
)
(
1 0
0 e-2iπ/3
)
(
e-iπ/3 0
0 e-iπ/3
)
(
-1 0
0 eiπ/3
)
(
-1 0
0 eiπ/3
)
6
(
0 1
-1 1
)
(
e-iπ/6 0
0 eiπ/6
)
6-
e-2iπ/3
eiπ/3
(
1 0
0 e2iπ/3
)
(
-1 0
0 e-iπ/3
)
(
e-iπ/6 0
0 e5iπ/6
)
(
-i 0
0 e-5iπ/6
)
(
i 0
0 eiπ/6
)
7
(
1 0
0 1
)
(
-1 0
0 -1
)
d1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
8
(
1 -1
1 0
)
(
e-5iπ/6 0
0 e5iπ/6
)
d6+
e2iπ/3
e-iπ/3
(
1 0
0 e-2iπ/3
)
(
-1 0
0 eiπ/3
)
(
e-5iπ/6 0
0 eiπ/6
)
(
-i 0
0 e-iπ/6
)
(
i 0
0 e5iπ/6
)
9
(
0 -1
1 -1
)
(
e-2iπ/3 0
0 e2iπ/3
)
d3+
e-2iπ/3
e-2iπ/3
(
1 0
0 e2iπ/3
)
(
1 0
0 e2iπ/3
)
(
e-2iπ/3 0
0 e-2iπ/3
)
(
1 0
0 e2iπ/3
)
(
1 0
0 e2iπ/3
)
10
(
-1 0
0 -1
)
(
i 0
0 -i
)
d2
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
i 0
0 -i
)
11
(
-1 1
-1 0
)
(
e2iπ/3 0
0 e-2iπ/3
)
d3-
e2iπ/3
e2iπ/3
(
1 0
0 e-2iπ/3
)
(
1 0
0 e-2iπ/3
)
(
e2iπ/3 0
0 e2iπ/3
)
(
1 0
0 e-2iπ/3
)
(
1 0
0 e-2iπ/3
)
12
(
0 1
-1 1
)
(
e5iπ/6 0
0 e-5iπ/6
)
d6-
e-2iπ/3
eiπ/3
(
1 0
0 e2iπ/3
)
(
-1 0
0 e-iπ/3
)
(
e5iπ/6 0
0 e-iπ/6
)
(
i 0
0 eiπ/6
)
(
-i 0
0 e-5iπ/6
)
13
(
1 0
0 1
)
(
1 0
0 1
)
1'
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
14
(
1 -1
1 0
)
(
eiπ/6 0
0 e-iπ/6
)
6+'
e-iπ/3
e2iπ/3
(
0 e2iπ/3
1 0
)
(
0 e-iπ/3
-1 0
)
(
0 e-iπ/6
e-iπ/6 0
)
(
0 eiπ/6
-i 0
)
(
0 e-5iπ/6
i 0
)
15
(
0 -1
1 -1
)
(
eiπ/3 0
0 e-iπ/3
)
3+'
eiπ/3
eiπ/3
(
0 e-2iπ/3
1 0
)
(
0 e-2iπ/3
1 0
)
(
0 e2iπ/3
e-iπ/3 0
)
(
0 e-2iπ/3
-1 0
)
(
0 e-2iπ/3
-1 0
)
16
(
-1 0
0 -1
)
(
-i 0
0 i
)
2'
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
17
(
-1 1
-1 0
)
(
e-iπ/3 0
0 eiπ/3
)
3-'
e-iπ/3
e-iπ/3
(
0 e2iπ/3
1 0
)
(
0 e2iπ/3
1 0
)
(
0 e-2iπ/3
eiπ/3 0
)
(
0 e2iπ/3
-1 0
)
(
0 e2iπ/3
-1 0
)
18
(
0 1
-1 1
)
(
e-iπ/6 0
0 eiπ/6
)
6-'
eiπ/3
e-2iπ/3
(
0 e-2iπ/3
1 0
)
(
0 eiπ/3
-1 0
)
(
0 eiπ/6
eiπ/6 0
)
(
0 e-iπ/6
i 0
)
(
0 e5iπ/6
-i 0
)
19
(
1 0
0 1
)
(
-1 0
0 -1
)
d1'
-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
1 0
)
(
e-5iπ/6 0
0 e5iπ/6
)
d6+'
e-iπ/3
e2iπ/3
(
0 e2iπ/3
1 0
)
(
0 e-iπ/3
-1 0
)
(
0 e5iπ/6
e5iπ/6 0
)
(
0 e-5iπ/6
i 0
)
(
0 eiπ/6
-i 0
)
21
(
0 -1
1 -1
)
(
e-2iπ/3 0
0 e2iπ/3
)
d3+'
eiπ/3
eiπ/3
(
0 e-2iπ/3
1 0
)
(
0 e-2iπ/3
1 0
)
(
0 e-iπ/3
e2iπ/3 0
)
(
0 eiπ/3
1 0
)
(
0 eiπ/3
1 0
)
22
(
-1 0
0 -1
)
(
i 0
0 -i
)
d2'
-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
-1 0
)
(
e2iπ/3 0
0 e-2iπ/3
)
d3-'
e-iπ/3
e-iπ/3
(
0 e2iπ/3
1 0
)
(
0 e2iπ/3
1 0
)
(
0 eiπ/3
e-2iπ/3 0
)
(
0 e-iπ/3
1 0
)
(
0 e-iπ/3
1 0
)
24
(
0 1
-1 1
)
(
e5iπ/6 0
0 e-5iπ/6
)
d6-'
eiπ/3
e-2iπ/3
(
0 e-2iπ/3
1 0
)
(
0 eiπ/3
-1 0
)
(
0 e-5iπ/6
e-5iπ/6 0
)
(
0 e5iπ/6
-i 0
)
(
0 e-iπ/6
i 0
)
k-Subgroupsmag
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