Bilbao Crystallographic Server Representations

## Irreducible representations of the Point Group 3m (No. 20)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 GM1+ A1g GM1+ 1 1 1 1 1 1 GM1- A1u GM1- 1 1 1 -1 -1 -1 GM2+ A2g GM2+ 1 1 -1 1 1 -1 GM2- A2u GM2- 1 1 -1 -1 -1 1 GM3+ Eg GM3+ 2 -1 0 2 -1 0 GM3- Eu GM3- 2 -1 0 -2 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: 3-001, 3+001 C3: 2120, 21-10, 2210 C4: -1 C5: -3-001, -3+001 C6: m120, m1-10, m210

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)
1
 ` 1 0 0 0 1 0 0 0 1`
1
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
2
 ` 0 -1 0 1 -1 0 0 0 1`
3+001
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
3
 ` -1 1 0 -1 0 0 0 0 1`
3-001
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
4
 ` 0 -1 0 -1 0 0 0 0 -1`
2110
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
5
 ` -1 1 0 0 1 0 0 0 -1`
2120
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
6
 ` 1 0 0 1 -1 0 0 0 -1`
2210
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
7
 ` -1 0 0 0 -1 0 0 0 -1`
1
 1
 -1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
8
 ` 0 1 0 -1 1 0 0 0 -1`
3+001
 1
 -1
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
9
 ` 1 -1 0 1 0 0 0 0 -1`
3-001
 1
 -1
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
10
 ` 0 1 0 1 0 0 0 0 1`
m110
 1
 -1
 -1
 1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
11
 ` 1 -1 0 0 -1 0 0 0 1`
m120
 1
 -1
 -1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
12
 ` -1 0 0 -1 1 0 0 0 1`
m210
 1
 -1
 -1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
k-Subgroupsmag