Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Point Group m3m (No. 32)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
-1
-1
-1
-1
-1
GM2+
A2g
GM2+
1
1
-1
1
-1
1
1
-1
1
-1
GM2-
A2u
GM2-
1
1
-1
1
-1
-1
-1
1
-1
1
GM3+
Eg
GM3+
2
2
0
-1
0
2
2
0
-1
0
GM3-
Eu
GM3-
2
2
0
-1
0
-2
-2
0
1
0
GM4+
T1g
GM4+
3
-1
-1
0
1
3
-1
-1
0
1
GM4-
T1u
GM4-
3
-1
-1
0
1
-3
1
1
0
-1
GM5+
T2g
GM5+
3
-1
1
0
-1
3
-1
1
0
-1
GM5-
T2u
GM5-
3
-1
1
0
-1
-3
1
-1
0
1
(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Notation of the irreps according to 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): Notation of the irreps according to 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 symmetry operations in the conjugacy classes

C1: 1
C2: 2001, 2010, 2100
C3: 201-1, 2011, 21-10, 2-101, 2101, 2110
C4: 3--11-1, 3-1-1-1, 3+1-1-1, 3+-1-11, 3+-11-1, 3+111, 3--1-11, 3-111
C5: 4+001, 4-010, 4+010, 4-001, 4+100, 4-100
C6: -1
C7: m001, m010, m100
C8: m01-1, m011, m1-10, m-101, m101, m110
C9: -3--11-1, -3-1-1-1, -3+1-1-1, -3+-1-11, -3+-11-1, -3+111, -3--1-11, -3-111
C10: -4+001, -4-010, -4+010, -4-001, -4+100, -4-100

Matrices of the representations of the group

The number in parentheses after the label of the irrep indicates the "reality" of the irrep: (1) for real, (-1) for pseudoreal and (0) for complex representations.

N
Matrix presentation
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5+(1)
GM5-(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
)
2
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
3
(
-1 0 0
0 1 0
0 0 -1
)
2010
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
4
(
1 0 0
0 -1 0
0 0 -1
)
2100
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
5
(
0 0 1
1 0 0
0 1 0
)
3+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
6
(
0 0 1
-1 0 0
0 -1 0
)
3+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
7
(
0 0 -1
-1 0 0
0 1 0
)
3+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
8
(
0 0 -1
1 0 0
0 -1 0
)
3+111
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
9
(
0 1 0
0 0 1
1 0 0
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
10
(
0 -1 0
0 0 1
-1 0 0
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
11
(
0 1 0
0 0 -1
-1 0 0
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
12
(
0 -1 0
0 0 -1
1 0 0
)
3-111
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
13
(
0 1 0
1 0 0
0 0 -1
)
2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 -1
0 -1 0
)
14
(
0 -1 0
-1 0 0
0 0 -1
)
2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 1
0 1 0
)
15
(
0 1 0
-1 0 0
0 0 1
)
4-001
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 -1
0 1 0
)
16
(
0 -1 0
1 0 0
0 0 1
)
4+001
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 1
0 -1 0
)
17
(
1 0 0
0 0 1
0 -1 0
)
4-100
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
18
(
-1 0 0
0 0 1
0 1 0
)
2011
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
19
(
-1 0 0
0 0 -1
0 -1 0
)
2011
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
20
(
1 0 0
0 0 -1
0 1 0
)
4+100
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
21
(
0 0 1
0 1 0
-1 0 0
)
4+010
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
22
(
0 0 1
0 -1 0
1 0 0
)
2101
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
23
(
0 0 -1
0 1 0
1 0 0
)
4-010
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
24
(
0 0 -1
0 -1 0
-1 0 0
)
2101
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
25
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0 0
0 1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 -1
)
26
(
1 0 0
0 1 0
0 0 -1
)
m001
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 1
)
27
(
1 0 0
0 -1 0
0 0 1
)
m010
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 -1
)
28
(
-1 0 0
0 1 0
0 0 1
)
m100
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 1
)
29
(
0 0 -1
-1 0 0
0 -1 0
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 -1 0
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 -1 0
)
30
(
0 0 -1
1 0 0
0 1 0
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 1 0
)
31
(
0 0 1
1 0 0
0 -1 0
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 1 0
)
32
(
0 0 1
-1 0 0
0 1 0
)
3+111
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 -1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 -1 0
)
33
(
0 -1 0
0 0 -1
-1 0 0
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 1
1 0 0
)
(
0 -1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
(
0 -1 0
0 0 -1
-1 0 0
)
34
(
0 1 0
0 0 -1
1 0 0
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 1 0
0 0 1
-1 0 0
)
35
(
0 -1 0
0 0 1
1 0 0
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 1 0
0 0 -1
1 0 0
)
36
(
0 1 0
0 0 1
-1 0 0
)
3-111
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 -1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 -1 0
0 0 1
1 0 0
)
37
(
0 -1 0
-1 0 0
0 0 1
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 -1
0 -1 0
)
(
-1 0 0
0 0 1
0 1 0
)
38
(
0 1 0
1 0 0
0 0 1
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 1
0 1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
39
(
0 -1 0
1 0 0
0 0 -1
)
4-001
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 -1
0 1 0
)
(
1 0 0
0 0 1
0 -1 0
)
40
(
0 1 0
-1 0 0
0 0 -1
)
4+001
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 1
0 -1 0
)
(
1 0 0
0 0 -1
0 1 0
)
41
(
-1 0 0
0 0 -1
0 1 0
)
4-100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
(
0 0 -1
0 1 0
1 0 0
)
42
(
1 0 0
0 0 -1
0 -1 0
)
m011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
(
0 0 1
0 -1 0
1 0 0
)
43
(
1 0 0
0 0 1
0 1 0
)
m011
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
44
(
-1 0 0
0 0 1
0 -1 0
)
4+100
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
(
0 0 1
0 1 0
-1 0 0
)
45
(
0 0 -1
0 -1 0
1 0 0
)
4+010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 1
)
46
(
0 0 -1
0 1 0
-1 0 0
)
m101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 -1
)
47
(
0 0 1
0 -1 0
-1 0 0
)
4-010
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 -1 0
1 0 0
0 0 -1
)
(
0 1 0
-1 0 0
0 0 1
)
48
(
0 0 1
0 1 0
1 0 0
)
m101
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 1 0
1 0 0
0 0 1
)
(
0 -1 0
-1 0 0
0 0 -1
)
k-Subgroupsmag
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