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Irreducible corepresentations of the Magnetic Point Group 4/m1' (N. 11.2.36)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
GM2+
Bg
GM2+
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
GM2-
Bu
GM2-
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
GM3+GM4+
1Eg2Eg
GM3+GM4+
2
-2
0
0
2
-2
0
0
2
-2
0
0
2
-2
0
0
GM3-GM4-
1Eu2Eu
GM3-GM4-
2
-2
0
0
-2
2
0
0
2
-2
0
0
-2
2
0
0
GM7+GM8+
1E2g2E2g
GM5GM7
2
0
-√2
-√2
2
0
-√2
-√2
-2
0
√2
√2
-2
0
√2
√2
GM5+GM6+
1E1g2E1g
GM6GM8
2
0
√2
√2
2
0
√2
√2
-2
0
-√2
-√2
-2
0
-√2
-√2
GM5-GM6-
1E1u2E1u
GM10GM12
2
0
√2
√2
-2
0
-√2
-√2
-2
0
-√2
-√2
2
0
√2
√2
GM8-GM7-
1E2u2E2u
GM11GM9
2
0
-√2
-√2
-2
0
√2
√2
-2
0
√2
√2
2
0
-√2
-√2
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: 2001
C3: 4+001
C4: 4-001
C5: 1
C6: m001
C7: 4+001
C8: 4-001
C9: d1
C10: d2001
C11: d4+001
C12: d4-001
C13: d1
C14: dm001
C15: d4+001
C16: d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM4+GM3-GM4-GM5GM7GM6GM8GM10GM12GM11GM9
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
2
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2001
1
1
1
1
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
3
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4+001
1
1
-1
-1
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
4
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4-001
1
1
-1
-1
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
5
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1
1
-1
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
6
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m001
1
-1
1
-1
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
7
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4+001
1
-1
-1
1
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
8
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4-001
1
-1
-1
1
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
9
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
10
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
1
1
1
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
11
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4+001
1
1
-1
-1
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
12
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4-001
1
1
-1
-1
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
13
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1
1
-1
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
14
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm001
1
-1
1
-1
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
15
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4+001
1
-1
-1
1
(
  i  0
  0 -i
)
(
 -i  0
  0  i
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
16
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4-001
1
-1
-1
1
(
 -i  0
  0  i
)
(
  i  0
  0 -i
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
17
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1'
1
-1
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
18
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2'001
1
-1
1
-1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
19
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4'+001
1
-1
-1
1
(
  0  i
 -i  0
)
(
  0 -i
  i  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
20
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4'-001
1
-1
-1
1
(
  0 -i
  i  0
)
(
  0  i
 -i  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
21
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1'
1
1
1
1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
22
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m'001
1
1
1
1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
23
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4'+001
1
1
-1
-1
(
  0  i
 -i  0
)
(
  0  i
 -i  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
24
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4'-001
1
1
-1
-1
(
  0 -i
  i  0
)
(
  0 -i
  i  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
25
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1'
1
-1
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
26
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2'001
1
-1
1
-1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
27
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4'+001
1
-1
-1
1
(
  0  i
 -i  0
)
(
  0 -i
  i  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
28
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4'-001
1
-1
-1
1
(
  0 -i
  i  0
)
(
  0  i
 -i  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
29
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1'
1
1
1
1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
30
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm'001
1
1
1
1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
31
(
  0  1  0
 -1  0  0
  0  0 -1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4'+001
1
1
-1
-1
(
  0  i
 -i  0
)
(
  0  i
 -i  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
32
(
  0 -1  0
  1  0  0
  0  0 -1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4'-001
1
1
-1
-1
(
  0 -i
  i  0
)
(
  0 -i
  i  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
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