Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Projective Magnetic Point Group 6πmm


Table of characters of the unitary symmetry operations


1
6+
3+
3-
2
6-
m11
m10
m01
m21
m12
dm1-1
m1-1
dm21
dm12
d1
d6+
d3+
d3-
d2
d6-
dm11
dm10
dm01
1E12E2
2
i√3
-1
0
(-i)√3
0
0
0
2
i√3
-1
0
(-i)√3
0
1E22E1
2
(-i)√3
-1
0
i√3
0
0
0
2
(-i)√3
-1
0
i√3
0
AB
2
0
2
0
0
0
0
0
2
0
2
0
0
0
1E1'
1
i
-1
i
-i
i
i
-i
-1
-i
1
-i
i
-i
1E1''
1
i
-1
i
-i
-i
-i
i
-1
-i
1
-i
i
i
2E1'
1
-i
-1
-i
i
i
-i
i
-1
i
1
i
-i
-i
2E1''
1
-i
-1
-i
i
-i
i
-i
-1
i
1
i
-i
i
1E21E3
2
-i
1
2i
i
0
0
0
-2
i
-1
-2i
-i
0
2E22E3
2
i
1
-2i
-i
0
0
0
-2
-i
-1
2i
i
0

Multiplication table of the symmetry operations


1
6+
3+
2
3-
6-
m11
m21
m10
m1-1
m01
m12
d1
d6+
d3+
d2
d3-
d6-
dm11
dm21
dm10
dm1-1
dm01
dm12
1
1
6+
3+
2
3-
6-
m11
m21
m10
m1-1
m01
m12
d1
d6+
d3+
d2
d3-
d6-
dm11
dm21
dm10
dm1-1
dm01
dm12
6+
6+
3+
d2
3-
6-
1
m12
dm11
dm21
m10
m1-1
dm01
d6+
d3+
2
d3-
d6-
d1
dm12
m11
m21
dm10
dm1-1
m01
3+
3+
d2
d3-
6-
1
6+
dm01
dm12
m11
dm21
m10
dm1-1
d3+
2
3-
d6-
d1
d6+
m01
m12
dm11
m21
dm10
m1-1
2
2
3-
6-
d1
d6+
d3+
m1-1
dm01
dm12
dm11
m21
m10
d2
d3-
d6-
1
6+
3+
dm1-1
m01
m12
m11
dm21
dm10
3-
3-
6-
1
d6+
d3+
2
m10
dm1-1
m01
dm12
dm11
dm21
d3-
d6-
d1
6+
3+
d2
dm10
m1-1
dm01
m12
m11
m21
6-
6-
1
6+
d3+
2
3-
dm21
dm10
m1-1
m01
dm12
m11
d6-
d1
d6+
3+
d2
d3-
m21
m10
dm1-1
dm01
m12
dm11
m11
m11
dm21
m10
dm1-1
dm01
m12
d1
6+
d3+
2
3-
d6-
dm11
m21
dm10
m1-1
m01
dm12
1
d6+
3+
d2
d3-
6-
m21
m21
dm10
dm1-1
m01
dm12
dm11
6-
d1
6+
3+
d2
3-
dm21
m10
m1-1
dm01
m12
m11
d6-
1
d6+
d3+
2
d3-
m10
m10
m1-1
m01
m12
m11
dm21
d3-
6-
d1
d6+
d3+
d2
dm10
dm1-1
dm01
dm12
dm11
m21
3-
d6-
1
6+
3+
2
m1-1
m1-1
m01
dm12
m11
dm21
m10
d2
3-
d6-
d1
d6+
3+
dm1-1
dm01
m12
dm11
m21
dm10
2
d3-
6-
1
6+
d3+
m01
m01
dm12
dm11
dm21
m10
m1-1
3+
2
d3-
d6-
d1
6+
dm01
m12
m11
m21
dm10
dm1-1
d3+
d2
3-
6-
1
d6+
m12
m12
m11
dm21
dm10
dm1-1
dm01
d6+
3+
2
3-
6-
d1
dm12
dm11
m21
m10
m1-1
m01
6+
d3+
d2
d3-
d6-
1
d1
d1
d6+
d3+
d2
d3-
d6-
dm11
dm21
dm10
dm1-1
dm01
dm12
1
6+
3+
2
3-
6-
m11
m21
m10
m1-1
m01
m12
d6+
d6+
d3+
2
d3-
d6-
d1
dm12
m11
m21
dm10
dm1-1
m01
6+
3+
d2
3-
6-
1
m12
dm11
dm21
m10
m1-1
dm01
d3+
d3+
2
3-
d6-
d1
d6+
m01
m12
dm11
m21
dm10
m1-1
3+
d2
d3-
6-
1
6+
dm01
dm12
m11
dm21
m10
dm1-1
d2
d2
d3-
d6-
1
6+
3+
dm1-1
m01
m12
m11
dm21
dm10
2
3-
6-
d1
d6+
d3+
m1-1
dm01
dm12
dm11
m21
m10
d3-
d3-
d6-
d1
6+
3+
d2
dm10
m1-1
dm01
m12
m11
m21
3-
6-
1
d6+
d3+
2
m10
dm1-1
m01
dm12
dm11
dm21
d6-
d6-
d1
d6+
3+
d2
d3-
m21
m10
dm1-1
dm01
m12
dm11
6-
1
6+
d3+
2
3-
dm21
dm10
m1-1
m01
dm12
m11
dm11
dm11
m21
dm10
m1-1
m01
dm12
1
d6+
3+
d2
d3-
6-
m11
dm21
m10
dm1-1
dm01
m12
d1
6+
d3+
2
3-
d6-
dm21
dm21
m10
m1-1
dm01
m12
m11
d6-
1
d6+
d3+
2
d3-
m21
dm10
dm1-1
m01
dm12
dm11
6-
d1
6+
3+
d2
3-
dm10
dm10
dm1-1
dm01
dm12
dm11
m21
3-
d6-
1
6+
3+
2
m10
m1-1
m01
m12
m11
dm21
d3-
6-
d1
d6+
d3+
d2
dm1-1
dm1-1
dm01
m12
dm11
m21
dm10
2
d3-
6-
1
6+
d3+
m1-1
m01
dm12
m11
dm21
m10
d2
3-
d6-
d1
d6+
3+
dm01
dm01
m12
m11
m21
dm10
dm1-1
d3+
d2
3-
6-
1
d6+
m01
dm12
dm11
dm21
m10
m1-1
3+
2
d3-
d6-
d1
6+
dm12
dm12
dm11
m21
m10
m1-1
m01
6+
d3+
d2
d3-
d6-
1
m12
m11
dm21
dm10
dm1-1
dm01
d6+
3+
2
3-
6-
d1

Table of projective phases in group multiplication


1
6+
3+
2
3-
6-
m11
m21
m10
m1-1
m01
m12
d1
d6+
d3+
d2
d3-
d6-
dm11
dm21
dm10
dm1-1
dm01
dm12
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
i
-i
-i
-i
-i
-i
1
1
1
1
1
1
i
-i
-i
-i
-i
-i
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
-i
-i
-i
i
i
i
1
1
1
1
1
1
-i
-i
-i
i
i
i
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
i
i
i
i
i
-i
1
1
1
1
1
1
i
i
i
i
i
-i
m11
1
-i
-1
i
1
-i
1
i
-1
-i
1
i
1
-i
-1
i
1
-i
1
i
-1
-i
1
i
m21
1
-i
-1
i
1
i
-i
1
i
-1
-i
1
1
-i
-1
i
1
i
-i
1
i
-1
-i
1
m10
1
-i
-1
i
-1
i
-1
-i
1
i
-1
-i
1
-i
-1
i
-1
i
-1
-i
1
i
-1
-i
m1-1
1
-i
-1
-i
-1
i
i
-1
-i
1
i
-1
1
-i
-1
-i
-1
i
i
-1
-i
1
i
-1
m01
1
-i
1
-i
-1
i
1
i
-1
-i
1
i
1
-i
1
-i
-1
i
1
i
-1
-i
1
i
m12
1
i
1
-i
-1
i
-i
1
i
-1
-i
1
1
i
1
-i
-1
i
-i
1
i
-1
-i
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
i
-i
-i
-i
-i
-i
1
1
1
1
1
1
i
-i
-i
-i
-i
-i
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
-i
-i
-i
i
i
i
1
1
1
1
1
1
-i
-i
-i
i
i
i
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
i
i
i
i
i
-i
1
1
1
1
1
1
i
i
i
i
i
-i
dm11
1
-i
-1
i
1
-i
1
i
-1
-i
1
i
1
-i
-1
i
1
-i
1
i
-1
-i
1
i
dm21
1
-i
-1
i
1
i
-i
1
i
-1
-i
1
1
-i
-1
i
1
i
-i
1
i
-1
-i
1
dm10
1
-i
-1
i
-1
i
-1
-i
1
i
-1
-i
1
-i
-1
i
-1
i
-1
-i
1
i
-1
-i
dm1-1
1
-i
-1
-i
-1
i
i
-1
-i
1
i
-1
1
-i
-1
-i
-1
i
i
-1
-i
1
i
-1
dm01
1
-i
1
-i
-1
i
1
i
-1
-i
1
i
1
-i
1
-i
-1
i
1
i
-1
-i
1
i
dm12
1
i
1
-i
-1
i
-i
1
i
-1
-i
1
1
i
1
-i
-1
i
-i
1
i
-1
-i
1

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol1E12E21E22E1AB1E1'1E1''2E1'2E1''1E21E32E22E3
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
2
(
  1 -1
  1  0
)
(
  eiπ/6  0
  0 e-iπ/6
)
6+
(
 e2iπ/3  0
  0  eiπ/3
)
(
  e-iπ/3  0
  0 e-2iπ/3
)
(
 -1  0
  0  1
)
i
i
-i
-i
(
 e-5iπ/6  0
  0  e-iπ/6
)
(
  eiπ/6  0
  0 e5iπ/6
)
3
(
  0 -1
  1 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
3+
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 1 0
 0 1
)
-1
-1
-1
-1
(
  eiπ/3  0
  0 e-iπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
4
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
2
(
  1  0
  0 -1
)
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
i
i
-i
-i
(
 i 0
 0 i
)
(
 -i  0
  0 -i
)
5
(
 -1  1
 -1  0
)
(
 e-iπ/3  0
  0  eiπ/3
)
3-
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 1 0
 0 1
)
-1
-1
-1
-1
(
 e-iπ/3  0
  0  eiπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
6
(
  0  1
 -1  1
)
(
 e-iπ/6  0
  0  eiπ/6
)
6-
(
 e-2iπ/3  0
  0  e-iπ/3
)
(
  eiπ/3  0
  0 e2iπ/3
)
(
 -1  0
  0  1
)
-i
-i
i
i
(
 e5iπ/6  0
  0  eiπ/6
)
(
  e-iπ/6  0
  0 e-5iπ/6
)
7
(
  0 -1
 -1  0
)
(
  0 e-2iπ/3
  e-iπ/3  0
)
m11
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
 0 1
 1 0
)
i
-i
i
-i
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
8
(
 -1  0
 -1  1
)
(
  0  eiπ/6
 e5iπ/6  0
)
m21
(
  0  e5iπ/6
 e-5iπ/6  0
)
(
  0 e-iπ/6
  eiπ/6  0
)
(
  0  i
 -i  0
)
i
-i
-i
i
(
  0  eiπ/3
 e2iπ/3  0
)
(
  0 e-2iπ/3
  e-iπ/3  0
)
9
(
 -1  1
  0  1
)
(
  0 -1
  1  0
)
m10
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 -1
 -1  0
)
i
-i
i
-i
(
  0  e-iπ/3
 e-2iπ/3  0
)
(
  0  e-iπ/3
 e-2iπ/3  0
)
10
(
 0 1
 1 0
)
(
  0 e5iπ/6
  eiπ/6  0
)
m1-1
(
  0  i
 -i  0
)
(
  0 -i
  i  0
)
(
  0 -i
  i  0
)
-i
i
i
-i
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
11
(
  1  0
  1 -1
)
(
  0 e2iπ/3
  eiπ/3  0
)
m01
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
 0 1
 1 0
)
i
-i
i
-i
(
  0  eiπ/3
 e2iπ/3  0
)
(
  0  eiπ/3
 e2iπ/3  0
)
12
(
  1 -1
  0 -1
)
(
  0 -i
 -i  0
)
m12
(
  0  eiπ/6
 e-iπ/6  0
)
(
  0 e-5iπ/6
  e5iπ/6  0
)
(
  0  i
 -i  0
)
i
-i
-i
i
(
  0  e-iπ/3
 e-2iπ/3  0
)
(
  0 e2iπ/3
  eiπ/3  0
)
13
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
d1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
-1
-1
-1
-1
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
14
(
  1 -1
  1  0
)
(
 e-5iπ/6  0
  0  e5iπ/6
)
d6+
(
 e2iπ/3  0
  0  eiπ/3
)
(
  e-iπ/3  0
  0 e-2iπ/3
)
(
 -1  0
  0  1
)
-i
-i
i
i
(
  eiπ/6  0
  0 e5iπ/6
)
(
 e-5iπ/6  0
  0  e-iπ/6
)
15
(
  0 -1
  1 -1
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
d3+
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 1 0
 0 1
)
1
1
1
1
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
16
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
d2
(
  1  0
  0 -1
)
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
-i
-i
i
i
(
 -i  0
  0 -i
)
(
 i 0
 0 i
)
17
(
 -1  1
 -1  0
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
d3-
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 1 0
 0 1
)
1
1
1
1
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
18
(
  0  1
 -1  1
)
(
  e5iπ/6  0
  0 e-5iπ/6
)
d6-
(
 e-2iπ/3  0
  0  e-iπ/3
)
(
  eiπ/3  0
  0 e2iπ/3
)
(
 -1  0
  0  1
)
i
i
-i
-i
(
  e-iπ/6  0
  0 e-5iπ/6
)
(
 e5iπ/6  0
  0  eiπ/6
)
19
(
  0 -1
 -1  0
)
(
  0  eiπ/3
 e2iπ/3  0
)
dm11
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
 0 1
 1 0
)
-i
i
-i
i
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
20
(
 -1  0
 -1  1
)
(
  0 e-5iπ/6
  e-iπ/6  0
)
dm21
(
  0  e5iπ/6
 e-5iπ/6  0
)
(
  0 e-iπ/6
  eiπ/6  0
)
(
  0  i
 -i  0
)
-i
i
i
-i
(
  0 e-2iπ/3
  e-iπ/3  0
)
(
  0  eiπ/3
 e2iπ/3  0
)
21
(
 -1  1
  0  1
)
(
  0  1
 -1  0
)
dm10
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 -1
 -1  0
)
-i
i
-i
i
(
  0 e2iπ/3
  eiπ/3  0
)
(
  0 e2iπ/3
  eiπ/3  0
)
22
(
 0 1
 1 0
)
(
  0  e-iπ/6
 e-5iπ/6  0
)
dm1-1
(
  0  i
 -i  0
)
(
  0 -i
  i  0
)
(
  0 -i
  i  0
)
i
-i
-i
i
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
23
(
  1  0
  1 -1
)
(
  0  e-iπ/3
 e-2iπ/3  0
)
dm01
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
 0 1
 1 0
)
-i
i
-i
i
(
  0 e-2iπ/3
  e-iπ/3  0
)
(
  0 e-2iπ/3
  e-iπ/3  0
)
24
(
  1 -1
  0 -1
)
(
 0 i
 i 0
)
dm12
(
  0  eiπ/6
 e-iπ/6  0
)
(
  0 e-5iπ/6
  e5iπ/6  0
)
(
  0  i
 -i  0
)
-i
i
i
-i
(
  0 e2iπ/3
  eiπ/3  0
)
(
  0  e-iπ/3
 e-2iπ/3  0
)
k-Subgroupsmag
Bilbao Crystallographic Server
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