Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 4/mmm1' (N. 15.2.54)


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
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
-1
-1
-1
-1
-1
1
1
-1
-1
GM3+
B1g
GM2+
1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
GM3-
B1u
GM2-
1
1
-1
1
-1
-1
-1
1
-1
1
1
-1
-1
1
GM2+
A2g
GM3+
1
1
1
-1
-1
1
1
1
-1
-1
1
1
1
1
GM2-
A2u
GM3-
1
1
1
-1
-1
-1
-1
-1
1
1
1
1
-1
-1
GM4+
B2g
GM4+
1
1
-1
-1
1
1
1
-1
-1
1
1
-1
1
-1
GM4-
B2u
GM4-
1
1
-1
-1
1
-1
-1
1
1
-1
1
-1
-1
1
GM5+
Eg
GM5+
2
-2
0
0
0
2
-2
0
0
0
2
0
2
0
GM5-
Eu
GM5-
2
-2
0
0
0
-2
2
0
0
0
2
0
-2
0
GM7+
E2g
GM6
2
0
-√2
0
0
2
0
-√2
0
0
-2
√2
-2
√2
GM6+
E1g
GM7
2
0
√2
0
0
2
0
√2
0
0
-2
-√2
-2
-√2
GM7-
E2u
GM8
2
0
-√2
0
0
-2
0
√2
0
0
-2
√2
2
-√2
GM6-
E1u
GM9
2
0
√2
0
0
-2
0
-√2
0
0
-2
-√2
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, d2001
C3: 4+001, 4-001
C4: 2010, 2100, d2010, d2100
C5: 2110, 2110, d2110, d2110
C6: 1
C7: m001, dm001
C8: 4+001, 4-001
C9: m010, m100, dm010, dm100
C10: m110, m110, dm110, dm110
C11: d1
C12: d4+001, d4-001
C13: d1
C14: d4+001, d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM3-GM4+GM4-GM5+GM5-GM6GM7GM8GM9
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
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
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
1
1
-1
-1
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/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
1
1
-1
-1
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
5
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0 -1
  1  0
)
2010
1
1
1
1
-1
-1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
6
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
2100
1
1
1
1
-1
-1
-1
-1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
7
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
2110
1
1
-1
-1
-1
-1
1
1
(
  1  0
  0 -1
)
(
  1  0
  0 -1
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
8
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
21-10
1
1
-1
-1
-1
-1
1
1
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
9
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1
1
-1
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
)
10
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m001
1
-1
1
-1
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
)
4+001
1
-1
-1
1
1
-1
-1
1
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
12
(
  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
1
-1
-1
1
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
13
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
m010
1
-1
1
-1
-1
1
-1
1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
14
(
 -1  0  0
  0  1  0
  0  0  1
)
(
  0 -i
 -i  0
)
m100
1
-1
1
-1
-1
1
-1
1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
15
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
m110
1
-1
-1
1
-1
1
1
-1
(
  1  0
  0 -1
)
(
 -1  0
  0  1
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
16
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
m1-10
1
-1
-1
1
-1
1
1
-1
(
 -1  0
  0  1
)
(
  1  0
  0 -1
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
17
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
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
)
18
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
1
1
1
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
)
19
(
  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
1
1
-1
-1
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
20
(
  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
1
1
-1
-1
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
21
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2010
1
1
1
1
-1
-1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
22
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
d2100
1
1
1
1
-1
-1
-1
-1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
23
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
d2110
1
1
-1
-1
-1
-1
1
1
(
  1  0
  0 -1
)
(
  1  0
  0 -1
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
24
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
d21-10
1
1
-1
-1
-1
-1
1
1
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
25
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1
1
-1
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
)
26
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm001
1
-1
1
-1
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
)
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
1
-1
-1
1
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
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
1
-1
-1
1
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
29
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
dm010
1
-1
1
-1
-1
1
-1
1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
30
(
 -1  0  0
  0  1  0
  0  0  1
)
(
 0 i
 i 0
)
dm100
1
-1
1
-1
-1
1
-1
1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
31
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
dm110
1
-1
-1
1
-1
1
1
-1
(
  1  0
  0 -1
)
(
 -1  0
  0  1
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
32
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
dm1-10
1
-1
-1
1
-1
1
1
-1
(
 -1  0
  0  1
)
(
  1  0
  0 -1
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
33
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1'
1
-1
1
-1
1
-1
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
34
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2'001
1
-1
1
-1
1
-1
1
-1
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
35
(
  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
1
-1
-1
1
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
36
(
  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
1
-1
-1
1
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
37
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0 -1
  1  0
)
2'010
1
-1
1
-1
-1
1
-1
1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
38
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
2'100
1
-1
1
-1
-1
1
-1
1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
39
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
2'110
1
-1
-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
)
40
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
2'1-10
1
-1
-1
1
-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
)
41
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1'
1
1
1
1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
42
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m'001
1
1
1
1
1
1
1
1
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
43
(
  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
1
1
-1
-1
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
44
(
  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
1
1
-1
-1
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
45
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
m'010
1
1
1
1
-1
-1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
46
(
 -1  0  0
  0  1  0
  0  0  1
)
(
  0 -i
 -i  0
)
m'100
1
1
1
1
-1
-1
-1
-1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
47
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
m'110
1
1
-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
)
48
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
m'1-10
1
1
-1
-1
-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
)
49
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1'
1
-1
1
-1
1
-1
1
-1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
50
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2'001
1
-1
1
-1
1
-1
1
-1
(
 -1  0
  0 -1
)
(
 1 0
 0 1
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
51
(
  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
1
-1
-1
1
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
52
(
  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
1
-1
-1
1
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
53
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2'010
1
-1
1
-1
-1
1
-1
1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
54
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
d2'100
1
-1
1
-1
-1
1
-1
1
(
  0 -1
 -1  0
)
(
 0 1
 1 0
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
55
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
d2'110
1
-1
-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
)
56
(
  0 -1  0
 -1  0  0
  0  0 -1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
d2'1-10
1
-1
-1
1
-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
)
57
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1'
1
1
1
1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
58
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm'001
1
1
1
1
1
1
1
1
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
59
(
  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
1
1
-1
-1
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-i3π/4  0
)
(
  0 ei3π/4
  eiπ/4  0
)
60
(
  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
1
1
-1
-1
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
(
  0  eiπ/4
 ei3π/4  0
)
(
  0 e-i3π/4
  e-iπ/4  0
)
61
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
dm'010
1
1
1
1
-1
-1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  ei3π/4  0
  0 e-i3π/4
)
62
(
 -1  0  0
  0  1  0
  0  0  1
)
(
 0 i
 i 0
)
dm'100
1
1
1
1
-1
-1
-1
-1
(
  0 -1
 -1  0
)
(
  0 -1
 -1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-i3π/4  0
  0  ei3π/4
)
63
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
dm'110
1
1
-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
)
64
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
dm'1-10
1
1
-1
-1
-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
)
k-Subgroupsmag
Bilbao Crystallographic Server
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