Bilbao Crystallographic Server Representations

## Irreducible representations of the Point Group 43m (No. 31)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 GM1 A1 GM1 1 1 1 1 1 GM2 A2 GM2 1 1 -1 -1 1 GM3 E GM3 2 2 0 0 -1 GM5 T2 GM4 3 -1 -1 1 0 GM4 T1 GM5 3 -1 1 -1 0
 (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: -4-100, -4+100, -4-001, -4+010, -4-010, -4+001 C4: m110, m101, m-101, m1-10, m011, m01-1 C5: 3--11-1, 3-1-1-1, 3+1-1-1, 3+-1-11, 3+-11-1, 3+111, 3--1-11, 3-111

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)
GM2(1)
GM3(1)
GM4(1)
GM5(1)
1
 ` 1 0 0 0 1 0 0 0 1`
1
 1
 1
 ` 1 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 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 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 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
 ` 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`
6
 ` 0 0 1 -1 0 0 0 -1 0`
3+111
 1
 1
 ` 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`
7
 ` 0 0 -1 -1 0 0 0 1 0`
3+111
 1
 1
 ` 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`
8
 ` 0 0 -1 1 0 0 0 -1 0`
3+111
 1
 1
 ` 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`
9
 ` 0 1 0 0 0 1 1 0 0`
3-111
 1
 1
 ` 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`
10
 ` 0 -1 0 0 0 1 -1 0 0`
3-111
 1
 1
 ` 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`
11
 ` 0 1 0 0 0 -1 -1 0 0`
3-111
 1
 1
 ` 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`
12
 ` 0 -1 0 0 0 -1 1 0 0`
3-111
 1
 1
 ` 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`
13
 ` 0 1 0 1 0 0 0 0 1`
m110
 1
 -1
 ` 0 1 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`
m110
 1
 -1
 ` 0 1 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
 ` 0 1 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
 ` 0 1 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`
m011
 1
 -1
 ` 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`
18
 ` -1 0 0 0 0 1 0 -1 0`
4+100
 1
 -1
 ` 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`
19
 ` -1 0 0 0 0 -1 0 1 0`
4-100
 1
 -1
 ` 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`
20
 ` 1 0 0 0 0 -1 0 -1 0`
m011
 1
 -1
 ` 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`
21
 ` 0 0 1 0 1 0 1 0 0`
m101
 1
 -1
 ` 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`
22
 ` 0 0 1 0 -1 0 -1 0 0`
4-010
 1
 -1
 ` 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`
23
 ` 0 0 -1 0 1 0 -1 0 0`
m101
 1
 -1
 ` 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`
24
 ` 0 0 -1 0 -1 0 1 0 0`
4+010
 1
 -1
 ` 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`
k-Subgroupsmag