Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Projective Magnetic Point Group 61'


Table of characters of the unitary symmetry operations


1
6+
3+
2
3-
6-
d1
d6+
d3+
d2
d3-
d6-
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
2E11E1
2
-1
-1
2
-1
-1
2
-1
-1
2
-1
-1
2E21E2
2
1
-1
-2
-1
1
2
1
-1
-2
-1
1
1E22E2
2
-√3
1
0
1
-√3
-2
√3
-1
0
-1
√3
1E32E3
2
√3
1
0
1
√3
-2
-√3
-1
0
-1
-√3
2E11E1
2
0
-2
0
-2
0
-2
0
2
0
2
0

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
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
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

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolAB2E11E12E21E21E22E21E32E32E11E1
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+
1
-1
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 e-5iπ/6  0
  0  e5iπ/6
)
(
  eiπ/6  0
  0 e-iπ/6
)
(
 -i  0
  0  i
)
3
(
  0 -1
  1 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
3+
1
1
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 -1  0
  0 -1
)
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-
1
1
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 -1  0
  0 -1
)
6
(
  0  1
 -1  1
)
(
 e-iπ/6  0
  0  eiπ/6
)
6-
1
-1
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
  e5iπ/6  0
  0 e-5iπ/6
)
(
 e-iπ/6  0
  0  eiπ/6
)
(
  i  0
  0 -i
)
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+
1
-1
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
  eiπ/6  0
  0 e-iπ/6
)
(
 e-5iπ/6  0
  0  e5iπ/6
)
(
  i  0
  0 -i
)
9
(
  0 -1
  1 -1
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
d3+
1
1
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
 1 0
 0 1
)
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-
1
1
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
  e2iπ/3  0
  0 e-2iπ/3
)
(
 1 0
 0 1
)
12
(
  0  1
 -1  1
)
(
  e5iπ/6  0
  0 e-5iπ/6
)
d6-
1
-1
(
 e-2iπ/3  0
  0  e2iπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 e-iπ/6  0
  0  eiπ/6
)
(
  e5iπ/6  0
  0 e-5iπ/6
)
(
 -i  0
  0  i
)
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+'
-1
1
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0  eiπ/6
 e5iπ/6  0
)
(
  0 e-5iπ/6
  e-iπ/6  0
)
(
 0 i
 i 0
)
15
(
  0 -1
  1 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
3+'
-1
-1
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0 e-2iπ/3
  e-iπ/3  0
)
(
  0 e-2iπ/3
  e-iπ/3  0
)
(
  0  1
 -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-'
-1
-1
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0 e2iπ/3
  eiπ/3  0
)
(
  0 e2iπ/3
  eiπ/3  0
)
(
  0  1
 -1  0
)
18
(
  0  1
 -1  1
)
(
 e-iπ/6  0
  0  eiπ/6
)
6-'
-1
1
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0  e-iπ/6
 e-5iπ/6  0
)
(
  0 e5iπ/6
  eiπ/6  0
)
(
  0 -i
 -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+'
-1
1
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 e-5iπ/6
  e-iπ/6  0
)
(
  0  eiπ/6
 e5iπ/6  0
)
(
  0 -i
 -i  0
)
21
(
  0 -1
  1 -1
)
(
 e-2iπ/3  0
  0  e2iπ/3
)
d3+'
-1
-1
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0  eiπ/3
 e2iπ/3  0
)
(
  0  eiπ/3
 e2iπ/3  0
)
(
  0 -1
  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-'
-1
-1
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0  e2iπ/3
 e-2iπ/3  0
)
(
  0  e-iπ/3
 e-2iπ/3  0
)
(
  0  e-iπ/3
 e-2iπ/3  0
)
(
  0 -1
  1  0
)
24
(
  0  1
 -1  1
)
(
  e5iπ/6  0
  0 e-5iπ/6
)
d6-'
-1
1
(
  0 e-2iπ/3
  e2iπ/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0 e5iπ/6
  eiπ/6  0
)
(
  0  e-iπ/6
 e-5iπ/6  0
)
(
 0 i
 i 0
)
k-Subgroupsmag
Bilbao Crystallographic Server
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