Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Projective Magnetic Point Group 6π/3'mm'


Table of characters of the unitary symmetry operations


1
3+
3-
m11
m01
m10
d1
d3+
d3-
dm11
dm01
dm10
2E'2E''
2
2e2iπ/3
2e-2iπ/3
0
0
2
2e2iπ/3
2e-2iπ/3
0
0
1EA
2
e-iπ/3
eiπ/3
0
0
2
e-iπ/3
eiπ/3
0
0
1E11E2
2
2e-iπ/3
2eiπ/3
0
0
-2
2e2iπ/3
2e-2iπ/3
0
0
2EE
2
e2iπ/3
e-2iπ/3
0
0
-2
e-iπ/3
eiπ/3
0
0

Multiplication table of the symmetry operations


1
3+
3-
m11
m10
m01
d1
d3+
d3-
dm11
dm10
dm01
6+'
2'
6-'
m'21
m'1-1
m'12
d6+'
d2'
d6-'
dm'21
dm'1-1
dm'12
1
1
3+
3-
m11
m10
m01
d1
d3+
d3-
dm11
dm10
dm01
6+'
2'
6-'
m'21
m'1-1
m'12
d6+'
d2'
d6-'
dm'21
dm'1-1
dm'12
3+
3+
d3-
1
dm01
m11
m10
d3+
3-
d1
m01
dm11
dm10
d2'
6-'
6+'
dm'12
dm'21
dm'1-1
2'
d6-'
d6+'
m'12
m'21
m'1-1
3-
3-
1
d3+
m10
m01
dm11
d3-
d1
3+
dm10
dm01
m11
6-'
d6+'
2'
dm'1-1
dm'12
dm'21
d6-'
6+'
d2'
m'1-1
m'12
m'21
m11
m11
m10
dm01
d1
d3+
3-
dm11
dm10
m01
1
3+
d3-
dm'21
dm'1-1
m'12
6+'
2'
d6-'
m'21
m'1-1
dm'12
d6+'
d2'
6-'
m10
m10
m01
m11
d3-
d1
d3+
dm10
dm01
dm11
3-
1
3+
m'1-1
m'12
dm'21
6-'
d6+'
d2'
dm'1-1
dm'12
m'21
d6-'
6+'
2'
m01
m01
dm11
m10
3+
d3-
d1
dm01
m11
dm10
d3+
3-
1
dm'12
dm'21
m'1-1
2'
d6-'
6+'
m'12
m'21
dm'1-1
d2'
6-'
d6+'
d1
d1
d3+
d3-
dm11
dm10
dm01
1
3+
3-
m11
m10
m01
d6+'
d2'
d6-'
dm'21
dm'1-1
dm'12
6+'
2'
6-'
m'21
m'1-1
m'12
d3+
d3+
3-
d1
m01
dm11
dm10
3+
d3-
1
dm01
m11
m10
2'
d6-'
d6+'
m'12
m'21
m'1-1
d2'
6-'
6+'
dm'12
dm'21
dm'1-1
d3-
d3-
d1
3+
dm10
dm01
m11
3-
1
d3+
m10
m01
dm11
d6-'
6+'
d2'
m'1-1
m'12
m'21
6-'
d6+'
2'
dm'1-1
dm'12
dm'21
dm11
dm11
dm10
m01
1
3+
d3-
m11
m10
dm01
d1
d3+
3-
m'21
m'1-1
dm'12
d6+'
d2'
6-'
dm'21
dm'1-1
m'12
6+'
2'
d6-'
dm10
dm10
dm01
dm11
3-
1
3+
m10
m01
m11
d3-
d1
d3+
dm'1-1
dm'12
m'21
d6-'
6+'
2'
m'1-1
m'12
dm'21
6-'
d6+'
d2'
dm01
dm01
m11
dm10
d3+
3-
1
m01
dm11
m10
3+
d3-
d1
m'12
m'21
dm'1-1
d2'
6-'
d6+'
dm'12
dm'21
m'1-1
2'
d6-'
6+'
6+'
6+'
d2'
6-'
m'12
dm'21
m'1-1
d6+'
2'
d6-'
dm'12
m'21
dm'1-1
d3+
d3-
d1
m11
dm10
m01
3+
3-
1
dm11
m10
dm01
2'
2'
6-'
d6+'
m'1-1
dm'12
m'21
d2'
d6-'
6+'
dm'1-1
m'12
dm'21
d3-
1
3+
m01
m11
dm10
3-
d1
d3+
dm01
dm11
m10
6-'
6-'
6+'
2'
dm'21
m'1-1
dm'12
d6-'
d6+'
d2'
m'21
dm'1-1
m'12
d1
3+
d3-
m10
dm01
dm11
1
d3+
3-
dm10
m01
m11
m'21
m'21
dm'1-1
dm'12
6-'
6+'
d2'
dm'21
m'1-1
m'12
d6-'
d6+'
2'
m10
dm01
m11
1
d3+
d3-
dm10
m01
dm11
d1
3+
3-
m'1-1
m'1-1
dm'12
dm'21
d2'
d6-'
d6+'
dm'1-1
m'12
m'21
2'
6-'
6+'
dm01
dm11
dm10
d3-
1
d3+
m01
m11
m10
3-
d1
3+
m'12
m'12
dm'21
dm'1-1
d6+'
2'
6-'
dm'12
m'21
m'1-1
6+'
d2'
d6-'
dm11
m10
m01
d3+
d3-
1
m11
dm10
dm01
3+
3-
d1
d6+'
d6+'
2'
d6-'
dm'12
m'21
dm'1-1
6+'
d2'
6-'
m'12
dm'21
m'1-1
3+
3-
1
dm11
m10
dm01
d3+
d3-
d1
m11
dm10
m01
d2'
d2'
d6-'
6+'
dm'1-1
m'12
dm'21
2'
6-'
d6+'
m'1-1
dm'12
m'21
3-
d1
d3+
dm01
dm11
m10
d3-
1
3+
m01
m11
dm10
d6-'
d6-'
d6+'
d2'
m'21
dm'1-1
m'12
6-'
6+'
2'
dm'21
m'1-1
dm'12
1
d3+
3-
dm10
m01
m11
d1
3+
d3-
m10
dm01
dm11
dm'21
dm'21
m'1-1
m'12
d6-'
d6+'
2'
m'21
dm'1-1
dm'12
6-'
6+'
d2'
dm10
m01
dm11
d1
3+
3-
m10
dm01
m11
1
d3+
d3-
dm'1-1
dm'1-1
m'12
m'21
2'
6-'
6+'
m'1-1
dm'12
dm'21
d2'
d6-'
d6+'
m01
m11
m10
3-
d1
3+
dm01
dm11
dm10
d3-
1
d3+
dm'12
dm'12
m'21
m'1-1
6+'
d2'
d6-'
m'12
dm'21
dm'1-1
d6+'
2'
6-'
m11
dm10
dm01
3+
3-
d1
dm11
m10
m01
d3+
d3-
1

Table of projective phases in group multiplication


1
3+
3-
m11
m10
m01
d1
d3+
d3-
dm11
dm10
dm01
6+'
2'
6-'
m'21
m'1-1
m'12
d6+'
d2'
d6-'
dm'21
dm'1-1
dm'12
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
3+
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e2iπ/3
e-iπ/3
e-iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e2iπ/3
e-iπ/3
e-iπ/3
3-
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
eiπ/3
eiπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
eiπ/3
eiπ/3
e-2iπ/3
m11
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
eiπ/6
i
e5iπ/6
e-iπ/6
-i
e-5iπ/6
eiπ/6
i
e5iπ/6
e-iπ/6
-i
e-5iπ/6
m10
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
eiπ/6
i
e-iπ/6
eiπ/6
e-iπ/6
-i
eiπ/6
i
e-iπ/6
eiπ/6
e-iπ/6
-i
m01
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
eiπ/6
-i
e-iπ/6
i
eiπ/6
e-iπ/6
eiπ/6
-i
e-iπ/6
i
eiπ/6
e-iπ/6
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
d3+
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e2iπ/3
e-iπ/3
e-iπ/3
e-2iπ/3
e-2iπ/3
e-2iπ/3
e2iπ/3
e-iπ/3
e-iπ/3
d3-
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
eiπ/3
eiπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
e2iπ/3
eiπ/3
eiπ/3
e-2iπ/3
dm11
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
eiπ/6
i
e5iπ/6
e-iπ/6
-i
e-5iπ/6
eiπ/6
i
e5iπ/6
e-iπ/6
-i
e-5iπ/6
dm10
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
eiπ/6
i
e-iπ/6
eiπ/6
e-iπ/6
-i
eiπ/6
i
e-iπ/6
eiπ/6
e-iπ/6
-i
dm01
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
eiπ/6
-i
e-iπ/6
i
eiπ/6
e-iπ/6
eiπ/6
-i
e-iπ/6
i
eiπ/6
e-iπ/6
6+'
1
1
1
e-5iπ/6
eiπ/6
eiπ/6
1
1
1
e-5iπ/6
eiπ/6
eiπ/6
eiπ/3
eiπ/3
eiπ/3
eiπ/6
eiπ/6
eiπ/6
eiπ/3
eiπ/3
eiπ/3
eiπ/6
eiπ/6
eiπ/6
2'
1
1
1
-i
-i
i
1
1
1
-i
-i
i
-1
-1
-1
-i
i
i
-1
-1
-1
-i
i
i
6-'
1
1
1
e-iπ/6
e-iπ/6
e-iπ/6
1
1
1
e-iπ/6
e-iπ/6
e-iπ/6
e-iπ/3
e-iπ/3
e-iπ/3
e-iπ/6
e-iπ/6
e5iπ/6
e-iπ/3
e-iπ/3
e-iπ/3
e-iπ/6
e-iπ/6
e5iπ/6
m'21
1
eiπ/3
e2iπ/3
eiπ/6
e-iπ/6
-i
1
eiπ/3
e2iπ/3
eiπ/6
e-iπ/6
-i
e-iπ/6
-i
eiπ/6
1
eiπ/3
e2iπ/3
e-iπ/6
-i
eiπ/6
1
eiπ/3
e2iπ/3
m'1-1
1
eiπ/3
e-iπ/3
i
eiπ/6
e-iπ/6
1
eiπ/3
e-iπ/3
i
eiπ/6
e-iπ/6
e-iπ/6
i
eiπ/6
e-iπ/3
1
eiπ/3
e-iπ/6
i
eiπ/6
e-iπ/3
1
eiπ/3
m'12
1
e-2iπ/3
e-iπ/3
e5iπ/6
i
eiπ/6
1
e-2iπ/3
e-iπ/3
e5iπ/6
i
eiπ/6
e5iπ/6
i
eiπ/6
e-2iπ/3
e-iπ/3
1
e5iπ/6
i
eiπ/6
e-2iπ/3
e-iπ/3
1
d6+'
1
1
1
e-5iπ/6
eiπ/6
eiπ/6
1
1
1
e-5iπ/6
eiπ/6
eiπ/6
eiπ/3
eiπ/3
eiπ/3
eiπ/6
eiπ/6
eiπ/6
eiπ/3
eiπ/3
eiπ/3
eiπ/6
eiπ/6
eiπ/6
d2'
1
1
1
-i
-i
i
1
1
1
-i
-i
i
-1
-1
-1
-i
i
i
-1
-1
-1
-i
i
i
d6-'
1
1
1
e-iπ/6
e-iπ/6
e-iπ/6
1
1
1
e-iπ/6
e-iπ/6
e-iπ/6
e-iπ/3
e-iπ/3
e-iπ/3
e-iπ/6
e-iπ/6
e5iπ/6
e-iπ/3
e-iπ/3
e-iπ/3
e-iπ/6
e-iπ/6
e5iπ/6
dm'21
1
eiπ/3
e2iπ/3
eiπ/6
e-iπ/6
-i
1
eiπ/3
e2iπ/3
eiπ/6
e-iπ/6
-i
e-iπ/6
-i
eiπ/6
1
eiπ/3
e2iπ/3
e-iπ/6
-i
eiπ/6
1
eiπ/3
e2iπ/3
dm'1-1
1
eiπ/3
e-iπ/3
i
eiπ/6
e-iπ/6
1
eiπ/3
e-iπ/3
i
eiπ/6
e-iπ/6
e-iπ/6
i
eiπ/6
e-iπ/3
1
eiπ/3
e-iπ/6
i
eiπ/6
e-iπ/3
1
eiπ/3
dm'12
1
e-2iπ/3
e-iπ/3
e5iπ/6
i
eiπ/6
1
e-2iπ/3
e-iπ/3
e5iπ/6
i
eiπ/6
e5iπ/6
i
eiπ/6
e-2iπ/3
e-iπ/3
1
e5iπ/6
i
eiπ/6
e-2iπ/3
e-iπ/3
1

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol2E'2E''1EA1E11E22EE
1
(
1 0
0 1
)
(
1 0
0 1
)
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
2
(
0 -1
1 -1
)
(
eiπ/3 0
0 e-iπ/3
)
3+
(
e2iπ/3 0
0 e2iπ/3
)
(
1 0
0 e-2iπ/3
)
(
e-iπ/3 0
0 e-iπ/3
)
(
-1 0
0 eiπ/3
)
3
(
-1 1
-1 0
)
(
e-iπ/3 0
0 eiπ/3
)
3-
(
e-2iπ/3 0
0 e-2iπ/3
)
(
1 0
0 e2iπ/3
)
(
eiπ/3 0
0 eiπ/3
)
(
-1 0
0 e-iπ/3
)
4
(
0 -1
-1 0
)
(
0 e-2iπ/3
e-iπ/3 0
)
m11
(
-1 0
0 1
)
(
0 1
1 0
)
(
-i 0
0 i
)
(
0 -1
1 0
)
5
(
-1 1
0 1
)
(
0 -1
1 0
)
m10
(
1 0
0 -1
)
(
0 e-iπ/3
eiπ/3 0
)
(
-i 0
0 i
)
(
0 e-iπ/3
e-2iπ/3 0
)
6
(
1 0
1 -1
)
(
0 e2iπ/3
eiπ/3 0
)
m01
(
-1 0
0 1
)
(
0 e-2iπ/3
e2iπ/3 0
)
(
-i 0
0 i
)
(
0 eiπ/3
e2iπ/3 0
)
7
(
1 0
0 1
)
(
-1 0
0 -1
)
d1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
8
(
0 -1
1 -1
)
(
e-2iπ/3 0
0 e2iπ/3
)
d3+
(
e2iπ/3 0
0 e2iπ/3
)
(
1 0
0 e-2iπ/3
)
(
e2iπ/3 0
0 e2iπ/3
)
(
1 0
0 e-2iπ/3
)
9
(
-1 1
-1 0
)
(
e2iπ/3 0
0 e-2iπ/3
)
d3-
(
e-2iπ/3 0
0 e-2iπ/3
)
(
1 0
0 e2iπ/3
)
(
e-2iπ/3 0
0 e-2iπ/3
)
(
1 0
0 e2iπ/3
)
10
(
0 -1
-1 0
)
(
0 eiπ/3
e2iπ/3 0
)
dm11
(
-1 0
0 1
)
(
0 1
1 0
)
(
i 0
0 -i
)
(
0 1
-1 0
)
11
(
-1 1
0 1
)
(
0 1
-1 0
)
dm10
(
1 0
0 -1
)
(
0 e-iπ/3
eiπ/3 0
)
(
i 0
0 -i
)
(
0 e2iπ/3
eiπ/3 0
)
12
(
1 0
1 -1
)
(
0 e-iπ/3
e-2iπ/3 0
)
dm01
(
-1 0
0 1
)
(
0 e-2iπ/3
e2iπ/3 0
)
(
i 0
0 -i
)
(
0 e-2iπ/3
e-iπ/3 0
)
13
(
1 -1
1 0
)
(
eiπ/6 0
0 e-iπ/6
)
6+'
(
0 e-iπ/3
e2iπ/3 0
)
(
0 1
e-iπ/3 0
)
(
0 e-iπ/3
e2iπ/3 0
)
(
0 1
e-iπ/3 0
)
14
(
-1 0
0 -1
)
(
-i 0
0 i
)
2'
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
15
(
0 1
-1 1
)
(
e-iπ/6 0
0 eiπ/6
)
6-'
(
0 eiπ/3
e-2iπ/3 0
)
(
0 1
eiπ/3 0
)
(
0 e-2iπ/3
eiπ/3 0
)
(
0 -1
e-2iπ/3 0
)
16
(
-1 0
-1 1
)
(
0 eiπ/6
e5iπ/6 0
)
m'21
(
0 i
i 0
)
(
e5iπ/6 0
0 e-5iπ/6
)
(
0 1
1 0
)
(
e-iπ/6 0
0 e-5iπ/6
)
17
(
0 1
1 0
)
(
0 e5iπ/6
eiπ/6 0
)
m'1-1
(
0 -i
-i 0
)
(
i 0
0 -i
)
(
0 -1
-1 0
)
(
-i 0
0 -i
)
18
(
1 -1
0 -1
)
(
0 -i
-i 0
)
m'12
(
0 i
i 0
)
(
eiπ/6 0
0 e-iπ/6
)
(
0 1
1 0
)
(
e-5iπ/6 0
0 e-iπ/6
)
19
(
1 -1
1 0
)
(
e-5iπ/6 0
0 e5iπ/6
)
d6+'
(
0 e-iπ/3
e2iπ/3 0
)
(
0 1
e-iπ/3 0
)
(
0 e2iπ/3
e-iπ/3 0
)
(
0 -1
e2iπ/3 0
)
20
(
-1 0
0 -1
)
(
i 0
0 -i
)
d2'
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
21
(
0 1
-1 1
)
(
e5iπ/6 0
0 e-5iπ/6
)
d6-'
(
0 eiπ/3
e-2iπ/3 0
)
(
0 1
eiπ/3 0
)
(
0 eiπ/3
e-2iπ/3 0
)
(
0 1
eiπ/3 0
)
22
(
-1 0
-1 1
)
(
0 e-5iπ/6
e-iπ/6 0
)
dm'21
(
0 i
i 0
)
(
e5iπ/6 0
0 e-5iπ/6
)
(
0 -1
-1 0
)
(
e5iπ/6 0
0 eiπ/6
)
23
(
0 1
1 0
)
(
0 e-iπ/6
e-5iπ/6 0
)
dm'1-1
(
0 -i
-i 0
)
(
i 0
0 -i
)
(
0 1
1 0
)
(
i 0
0 i
)
24
(
1 -1
0 -1
)
(
0 i
i 0
)
dm'12
(
0 i
i 0
)
(
eiπ/6 0
0 e-iπ/6
)
(
0 -1
-1 0
)
(
eiπ/6 0
0 e5iπ/6
)
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
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