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 Koster GF, Dimmok JO, Wheeler RG and Statz H, (1963) Properties of the thirty-two point groups, M.I.T. Press, Cambridge, Mass. (2): Notation of the irreps according to Mulliken RS (1933) Phys. Rev. 43, 279-302. (3): Notation of the irreps according to C. J. Bradley, A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) 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