Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 61' (N. 21.2.77)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1
A
GM1
1
1
1
1
1
1
1
1
1
1
1
1
GM4
B
GM2
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM5GM6
1E12E1
GM3GM5
2
-1
-1
2
-1
-1
2
-1
-1
2
-1
-1
GM2GM3
1E22E2
GM4GM6
2
-1
-1
-2
1
1
2
-1
-1
-2
1
1
GM12GM11
1E12E1
GM7GM8
2
-2
-2
0
0
0
-2
2
2
0
0
0
GM7GM8
1E32E3
GM10GM11
2
1
1
0
√3
√3
-2
-1
-1
0
-√3
-√3
GM10GM9
1E22E2
GM12GM9
2
1
1
0
-√3
-√3
-2
-1
-1
0
√3
√3
The notation used in this table is an extension to corepresentations of the following notations used for irreducible representations:
(1): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

Lists of unitary symmetry operations in the conjugacy classes

C1: 1
C2: 3+001
C3: 3-001
C4: 2001
C5: 6-001
C6: 6+001
C7: d1
C8: d3+001
C9: d3-001
C10: d2001
C11: d6-001
C12: d6+001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM5GM4GM6GM7GM8GM10GM11GM12GM9
1
(
 1 0 0
 0 1 0
 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
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3+001
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
3
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3-001
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
4
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2001
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
5
(
  0  1  0
 -1  1  0
  0  0  1
)
(
 (√3-i)/2  0
  0 (√3+i)/2
)
6-001
1
-1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
  i  0
  0 -i
)
(
  eiπ/6  0
  0 e-iπ/6
)
(
  ei5π/6  0
  0 e-i5π/6
)
6
(
  1 -1  0
  1  0  0
  0  0  1
)
(
 (√3+i)/2  0
  0 (√3-i)/2
)
6+001
1
-1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 -i  0
  0  i
)
(
 e-iπ/6  0
  0  eiπ/6
)
(
 e-i5π/6  0
  0  ei5π/6
)
7
(
 1 0 0
 0 1 0
 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
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3+001
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
9
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3-001
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
10
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
11
(
  0  1  0
 -1  1  0
  0  0  1
)
(
 -(√3-i)/2  0
  0 -(√3+i)/2
)
d6-001
1
-1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 -i  0
  0  i
)
(
 e-i5π/6  0
  0  ei5π/6
)
(
 e-iπ/6  0
  0  eiπ/6
)
12
(
  1 -1  0
  1  0  0
  0  0  1
)
(
 -(√3+i)/2  0
  0 -(√3-i)/2
)
d6+001
1
-1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
  i  0
  0 -i
)
(
  ei5π/6  0
  0 e-i5π/6
)
(
  eiπ/6  0
  0 e-iπ/6
)
13
(
 1 0 0
 0 1 0
 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
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3'+001
1
1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
15
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3'-001
1
1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
(
  0 ei2π/3
  eiπ/3  0
)
16
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2'001
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
17
(
  0  1  0
 -1  1  0
  0  0  1
)
(
 (√3-i)/2  0
  0 (√3+i)/2
)
6'-001
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 -i
 -i  0
)
(
  0 e-i5π/6
  e-iπ/6  0
)
(
  0  e-iπ/6
 e-i5π/6  0
)
18
(
  1 -1  0
  1  0  0
  0  0  1
)
(
 (√3+i)/2  0
  0 (√3-i)/2
)
6'+001
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
 0 i
 i 0
)
(
  0 ei5π/6
  eiπ/6  0
)
(
  0  eiπ/6
 ei5π/6  0
)
19
(
 1 0 0
 0 1 0
 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
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3'+001
1
1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
(
  0  eiπ/3
 ei2π/3  0
)
21
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3'-001
1
1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
22
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2'001
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
23
(
  0  1  0
 -1  1  0
  0  0  1
)
(
 -(√3-i)/2  0
  0 -(√3+i)/2
)
d6'-001
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
 0 i
 i 0
)
(
  0  eiπ/6
 ei5π/6  0
)
(
  0 ei5π/6
  eiπ/6  0
)
24
(
  1 -1  0
  1  0  0
  0  0  1
)
(
 -(√3+i)/2  0
  0 -(√3-i)/2
)
d6'+001
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0 -i
 -i  0
)
(
  0  e-iπ/6
 e-i5π/6  0
)
(
  0 e-i5π/6
  e-iπ/6  0
)
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