Bilbao Crystallographic Server Representations

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

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 GM1+ A1g GM1+ 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 GM2+ A2g GM2+ 1 1 -1 1 1 -1 1 1 -1 1 1 -1 GM2- A2u GM2- 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 GM3+ Eg GM3+ 2 -1 0 2 -1 0 2 -1 0 2 -1 0 GM3- Eu GM3- 2 -1 0 2 -1 0 -2 1 0 -2 1 0 GM4+ 2Eg GM4 1 -1 -i -1 1 i 1 -1 -i -1 1 i GM5+ 1Eg GM5 1 -1 i -1 1 -i 1 -1 i -1 1 -i GM4- 2Eu GM6 1 -1 -i -1 1 i -1 1 i 1 -1 -i GM5- 1Eu GM7 1 -1 i -1 1 -i -1 1 -i 1 -1 i GM6+ E1g GM8 2 1 0 -2 -1 0 2 1 0 -2 -1 0 GM6- E1u GM9 2 1 0 -2 -1 0 -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: 21-10, 2120, 2210 C4: d1 C5: d3+001, d3-001 C6: d21-10, d2120, d2210 C7: -1 C8: -3+001, -3-001 C9: m1-10, m120, m210 C10: d-1 C11: d-3+001, d-3-001 C12: dm1-10, dm120, dm210

List of pairs of conjugated irreducible representations

(*GM4,*GM5)
(*GM6,*GM7)
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(0)
GM5(0)
GM6(0)
GM7(0)
GM8(-1)
GM9(-1)
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
2
 ` 0 -1 0 1 -1 0 0 0 1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3+001
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 -1
 -1
 -1
 -1
 ` e-iπ/3 0 0 eiπ/3`
 ` e-iπ/3 0 0 eiπ/3`
3
 ` -1 1 0 -1 0 0 0 0 1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3-001
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 -1
 -1
 -1
 -1
 ` eiπ/3 0 0 e-iπ/3`
 ` eiπ/3 0 0 e-iπ/3`
4
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
2110
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 -i
 i
 -i
 i
 ` 0 -1 1 0`
 ` 0 -1 1 0`
5
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
2120
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 -i
 i
 -i
 i
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 eiπ/3 ei2π/3 0`
6
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
2210
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 -i
 i
 -i
 i
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 e-iπ/3 e-i2π/3 0`
7
 ` -1 0 0 0 -1 0 0 0 -1`
 ` 1 0 0 1`
1
 1
 -1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 1
 1
 -1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
8
 ` 0 1 0 -1 1 0 0 0 -1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3+001
 1
 -1
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 -1
 -1
 1
 1
 ` e-iπ/3 0 0 eiπ/3`
 ` ei2π/3 0 0 e-i2π/3`
9
 ` 1 -1 0 1 0 0 0 0 -1`
 ` (1-i√3)/2 0 0 (1+i√3)/2`
3-001
 1
 -1
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 -1
 -1
 1
 1
 ` eiπ/3 0 0 e-iπ/3`
 ` e-i2π/3 0 0 ei2π/3`
10
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 -(√3-i)/2 (√3+i)/2 0`
m110
 1
 -1
 -1
 1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 -i
 i
 i
 -i
 ` 0 -1 1 0`
 ` 0 1 -1 0`
11
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 -i -i 0`
m120
 1
 -1
 -1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 -i
 i
 i
 -i
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 e-i2π/3 e-iπ/3 0`
12
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 (√3+i)/2 -(√3-i)/2 0`
m210
 1
 -1
 -1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 -i
 i
 i
 -i
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 ei2π/3 eiπ/3 0`
13
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 1
 1
 ` 1 0 0 1`
 ` 1 0 0 1`
 -1
 -1
 -1
 -1
 ` -1 0 0 -1`
 ` -1 0 0 -1`
14
 ` 0 -1 0 1 -1 0 0 0 1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
 1
 1
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` ei2π/3 0 0 e-i2π/3`
15
 ` -1 1 0 -1 0 0 0 0 1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
 1
 1
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` e-i2π/3 0 0 ei2π/3`
16
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
d2110
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 i
 -i
 i
 -i
 ` 0 1 -1 0`
 ` 0 1 -1 0`
17
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 i i 0`
d2120
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 i
 -i
 i
 -i
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 e-i2π/3 e-iπ/3 0`
18
 ` 1 0 0 1 -1 0 0 0 -1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
d2210
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 i
 -i
 i
 -i
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 ei2π/3 eiπ/3 0`
19
 ` -1 0 0 0 -1 0 0 0 -1`
 ` -1 0 0 -1`
d1
 1
 -1
 1
 -1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 -1
 -1
 1
 1
 ` -1 0 0 -1`
 ` 1 0 0 1`
20
 ` 0 1 0 -1 1 0 0 0 -1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 -1
 1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 1
 1
 -1
 -1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
21
 ` 1 -1 0 1 0 0 0 0 -1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 -1
 1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 1
 1
 -1
 -1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
22
 ` 0 1 0 1 0 0 0 0 1`
 ` 0 (√3-i)/2 -(√3+i)/2 0`
dm110
 1
 -1
 -1
 1
 ` 0 1 1 0`
 ` 0 -1 -1 0`
 i
 -i
 -i
 i
 ` 0 1 -1 0`
 ` 0 -1 1 0`
23
 ` 1 -1 0 0 -1 0 0 0 1`
 ` 0 i i 0`
dm120
 1
 -1
 -1
 1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 eiπ/3 e-iπ/3 0`
 i
 -i
 -i
 i
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 eiπ/3 ei2π/3 0`
24
 ` -1 0 0 -1 1 0 0 0 1`
 ` 0 -(√3+i)/2 (√3-i)/2 0`
dm210
 1
 -1
 -1
 1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 e-iπ/3 eiπ/3 0`
 i
 -i
 -i
 i
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 e-iπ/3 e-i2π/3 0`
k-Subgroupsmag