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Irreducible corepresentations of the Projective Magnetic Point Group 6m'm'


Table of characters of the unitary symmetry operations


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
2E1
1
e-2iπ/3
e2iπ/3
1
e-2iπ/3
e2iπ/3
1
e-2iπ/3
e2iπ/3
1
e-2iπ/3
e2iπ/3
2E2
1
eiπ/3
e2iπ/3
-1
e-2iπ/3
e-iπ/3
1
eiπ/3
e2iπ/3
-1
e-2iπ/3
e-iπ/3
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
A
1
1
1
1
1
1
1
1
1
1
1
1
B
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1E3
1
e-iπ/6
e-iπ/3
i
eiπ/3
eiπ/6
-1
e5iπ/6
e2iπ/3
-i
e-2iπ/3
e-5iπ/6
2E2
1
e5iπ/6
e-iπ/3
-i
eiπ/3
e-5iπ/6
-1
e-iπ/6
e2iπ/3
i
e-2iπ/3
eiπ/6
1E2
1
e-5iπ/6
eiπ/3
i
e-iπ/3
e5iπ/6
-1
eiπ/6
e-2iπ/3
-i
e2iπ/3
e-iπ/6
2E3
1
eiπ/6
eiπ/3
-i
e-iπ/3
e-iπ/6
-1
e-5iπ/6
e-2iπ/3
i
e2iπ/3
e5iπ/6
1E1
1
i
-1
i
-1
-i
-1
-i
1
-i
1
i
2E1
1
-i
-1
-i
-1
i
-1
i
1
i
1
-i

Multiplication table of the symmetry operations


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
m'11
m'21
m'10
m'1-1
m'01
m'12
dm'11
dm'21
dm'10
dm'1-1
dm'01
dm'12
1
1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
m'11
m'21
m'10
m'1-1
m'01
m'12
dm'11
dm'21
dm'10
dm'1-1
dm'01
dm'12
6+
6+
3+
d2
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
m'12
dm'11
dm'21
m'10
m'1-1
dm'01
dm'12
m'11
m'21
dm'10
dm'1-1
m'01
3+
3+
d2
d3-
6-
1
6+
d3+
2
3-
d6-
d1
d6+
dm'01
dm'12
m'11
dm'21
m'10
dm'1-1
m'01
m'12
dm'11
m'21
dm'10
m'1-1
2
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
m'1-1
dm'01
dm'12
dm'11
m'21
m'10
dm'1-1
m'01
m'12
m'11
dm'21
dm'10
3-
3-
6-
1
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
m'10
dm'1-1
m'01
dm'12
dm'11
dm'21
dm'10
m'1-1
dm'01
m'12
m'11
m'21
6-
6-
1
6+
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-
dm'21
dm'10
m'1-1
m'01
dm'12
m'11
m'21
m'10
dm'1-1
dm'01
m'12
dm'11
d1
d1
d6+
d3+
d2
d3-
d6-
1
6+
3+
2
3-
6-
dm'11
dm'21
dm'10
dm'1-1
dm'01
dm'12
m'11
m'21
m'10
m'1-1
m'01
m'12
d6+
d6+
d3+
2
d3-
d6-
d1
6+
3+
d2
3-
6-
1
dm'12
m'11
m'21
dm'10
dm'1-1
m'01
m'12
dm'11
dm'21
m'10
m'1-1
dm'01
d3+
d3+
2
3-
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
m'01
m'12
dm'11
m'21
dm'10
m'1-1
dm'01
dm'12
m'11
dm'21
m'10
dm'1-1
d2
d2
d3-
d6-
1
6+
3+
2
3-
6-
d1
d6+
d3+
dm'1-1
m'01
m'12
m'11
dm'21
dm'10
m'1-1
dm'01
dm'12
dm'11
m'21
m'10
d3-
d3-
d6-
d1
6+
3+
d2
3-
6-
1
d6+
d3+
2
dm'10
m'1-1
dm'01
m'12
m'11
m'21
m'10
dm'1-1
m'01
dm'12
dm'11
dm'21
d6-
d6-
d1
d6+
3+
d2
d3-
6-
1
6+
d3+
2
3-
m'21
m'10
dm'1-1
dm'01
m'12
dm'11
dm'21
dm'10
m'1-1
m'01
dm'12
m'11
m'11
m'11
dm'21
m'10
dm'1-1
dm'01
m'12
dm'11
m'21
dm'10
m'1-1
m'01
dm'12
1
d6+
3+
d2
d3-
6-
d1
6+
d3+
2
3-
d6-
m'21
m'21
dm'10
dm'1-1
m'01
dm'12
dm'11
dm'21
m'10
m'1-1
dm'01
m'12
m'11
d6-
1
d6+
d3+
2
d3-
6-
d1
6+
3+
d2
3-
m'10
m'10
m'1-1
m'01
m'12
m'11
dm'21
dm'10
dm'1-1
dm'01
dm'12
dm'11
m'21
3-
d6-
1
6+
3+
2
d3-
6-
d1
d6+
d3+
d2
m'1-1
m'1-1
m'01
dm'12
m'11
dm'21
m'10
dm'1-1
dm'01
m'12
dm'11
m'21
dm'10
2
d3-
6-
1
6+
d3+
d2
3-
d6-
d1
d6+
3+
m'01
m'01
dm'12
dm'11
dm'21
m'10
m'1-1
dm'01
m'12
m'11
m'21
dm'10
dm'1-1
d3+
d2
3-
6-
1
d6+
3+
2
d3-
d6-
d1
6+
m'12
m'12
m'11
dm'21
dm'10
dm'1-1
dm'01
dm'12
dm'11
m'21
m'10
m'1-1
m'01
6+
d3+
d2
d3-
d6-
1
d6+
3+
2
3-
6-
d1
dm'11
dm'11
m'21
dm'10
m'1-1
m'01
dm'12
m'11
dm'21
m'10
dm'1-1
dm'01
m'12
d1
6+
d3+
2
3-
d6-
1
d6+
3+
d2
d3-
6-
dm'21
dm'21
m'10
m'1-1
dm'01
m'12
m'11
m'21
dm'10
dm'1-1
m'01
dm'12
dm'11
6-
d1
6+
3+
d2
3-
d6-
1
d6+
d3+
2
d3-
dm'10
dm'10
dm'1-1
dm'01
dm'12
dm'11
m'21
m'10
m'1-1
m'01
m'12
m'11
dm'21
d3-
6-
d1
d6+
d3+
d2
3-
d6-
1
6+
3+
2
dm'1-1
dm'1-1
dm'01
m'12
dm'11
m'21
dm'10
m'1-1
m'01
dm'12
m'11
dm'21
m'10
d2
3-
d6-
d1
d6+
3+
2
d3-
6-
1
6+
d3+
dm'01
dm'01
m'12
m'11
m'21
dm'10
dm'1-1
m'01
dm'12
dm'11
dm'21
m'10
m'1-1
3+
2
d3-
d6-
d1
6+
d3+
d2
3-
6-
1
d6+
dm'12
dm'12
dm'11
m'21
m'10
m'1-1
m'01
m'12
m'11
dm'21
dm'10
dm'1-1
dm'01
d6+
3+
2
3-
6-
d1
6+
d3+
d2
d3-
d6-
1

Table of projective phases in group multiplication


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
m'11
m'21
m'10
m'1-1
m'01
m'12
dm'11
dm'21
dm'10
dm'1-1
dm'01
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
6+
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
1
1
1
1
1
1
1
1
1
1
1
1
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
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
1
1
1
1
1
1
1
1
1
1
1
1
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
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
1
1
1
1
1
1
1
1
1
1
1
1
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
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
1
1
1
1
1
1
1
1
1
1
1
1
m'11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m'21
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m'10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m'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
1
m'01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m'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
dm'11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm'21
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm'10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm'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
1
dm'01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol2E12E21E11E2AB1E32E21E22E31E12E1
1
(
1 0
0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
1
1
1
1
1
2
(
1 -1
1 0
)
(
eiπ/6 0
0 e-iπ/6
)
6+
e-2iπ/3
eiπ/3
e2iπ/3
e-iπ/3
1
-1
e-iπ/6
e5iπ/6
e-5iπ/6
eiπ/6
i
-i
3
(
0 -1
1 -1
)
(
eiπ/3 0
0 e-iπ/3
)
3+
e2iπ/3
e2iπ/3
e-2iπ/3
e-2iπ/3
1
1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
-1
-1
4
(
-1 0
0 -1
)
(
-i 0
0 i
)
2
1
-1
1
-1
1
-1
i
-i
i
-i
i
-i
5
(
-1 1
-1 0
)
(
e-iπ/3 0
0 eiπ/3
)
3-
e-2iπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
1
1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
-1
-1
6
(
0 1
-1 1
)
(
e-iπ/6 0
0 eiπ/6
)
6-
e2iπ/3
e-iπ/3
e-2iπ/3
eiπ/3
1
-1
eiπ/6
e-5iπ/6
e5iπ/6
e-iπ/6
-i
i
7
(
1 0
0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
8
(
1 -1
1 0
)
(
e-5iπ/6 0
0 e5iπ/6
)
d6+
e-2iπ/3
eiπ/3
e2iπ/3
e-iπ/3
1
-1
e5iπ/6
e-iπ/6
eiπ/6
e-5iπ/6
-i
i
9
(
0 -1
1 -1
)
(
e-2iπ/3 0
0 e2iπ/3
)
d3+
e2iπ/3
e2iπ/3
e-2iπ/3
e-2iπ/3
1
1
e2iπ/3
e2iπ/3
e-2iπ/3
e-2iπ/3
1
1
10
(
-1 0
0 -1
)
(
i 0
0 -i
)
d2
1
-1
1
-1
1
-1
-i
i
-i
i
-i
i
11
(
-1 1
-1 0
)
(
e2iπ/3 0
0 e-2iπ/3
)
d3-
e-2iπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
1
1
e-2iπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
1
1
12
(
0 1
-1 1
)
(
e5iπ/6 0
0 e-5iπ/6
)
d6-
e2iπ/3
e-iπ/3
e-2iπ/3
eiπ/3
1
-1
e-5iπ/6
eiπ/6
e-iπ/6
e5iπ/6
i
-i
13
(
0 -1
-1 0
)
(
0 e-2iπ/3
e-iπ/3 0
)
m'11
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
14
(
-1 0
-1 1
)
(
0 eiπ/6
e5iπ/6 0
)
m'21
eiπ/3
e-2iπ/3
e-iπ/3
e2iπ/3
-1
1
e-iπ/6
e5iπ/6
e-5iπ/6
eiπ/6
i
-i
15
(
-1 1
0 1
)
(
0 -1
1 0
)
m'10
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
-1
-1
e2iπ/3
e2iπ/3
e-2iπ/3
e-2iπ/3
1
1
16
(
0 1
1 0
)
(
0 e5iπ/6
eiπ/6 0
)
m'1-1
-1
1
-1
1
-1
1
i
-i
i
-i
i
-i
17
(
1 0
1 -1
)
(
0 e2iπ/3
eiπ/3 0
)
m'01
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
-1
-1
18
(
1 -1
0 -1
)
(
0 -i
-i 0
)
m'12
e-iπ/3
e2iπ/3
eiπ/3
e-2iπ/3
-1
1
e-5iπ/6
eiπ/6
e-iπ/6
e5iπ/6
i
-i
19
(
0 -1
-1 0
)
(
0 eiπ/3
e2iπ/3 0
)
dm'11
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
20
(
-1 0
-1 1
)
(
0 e-5iπ/6
e-iπ/6 0
)
dm'21
eiπ/3
e-2iπ/3
e-iπ/3
e2iπ/3
-1
1
e5iπ/6
e-iπ/6
eiπ/6
e-5iπ/6
-i
i
21
(
-1 1
0 1
)
(
0 1
-1 0
)
dm'10
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
-1
-1
22
(
0 1
1 0
)
(
0 e-iπ/6
e-5iπ/6 0
)
dm'1-1
-1
1
-1
1
-1
1
-i
i
-i
i
-i
i
23
(
1 0
1 -1
)
(
0 e-iπ/3
e-2iπ/3 0
)
dm'01
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
-1
-1
e-2iπ/3
e-2iπ/3
e2iπ/3
e2iπ/3
1
1
24
(
1 -1
0 -1
)
(
0 i
i 0
)
dm'12
e-iπ/3
e2iπ/3
eiπ/3
e-2iπ/3
-1
1
eiπ/6
e-5iπ/6
e5iπ/6
e-iπ/6
-i
i
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
For comments, please mail to
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