Bilbao Crystallographic Server Representations

## Irreducible representations of the Double Point Group 32 (No. 18)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 GM1 A1 GM1 1 1 1 1 1 1 GM2 A2 GM2 1 1 -1 1 1 -1 GM3 E GM3 2 -1 0 2 -1 0 GM6 2E GM4 1 -1 -i -1 1 i GM5 1E GM5 1 -1 i -1 1 -i GM4 E1 GM6 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

List of pairs of conjugated irreducible representations

(*GM4,*GM5)
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(0)
GM5(0)
GM6(-1)
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 ` 1 0 0 1`
 1
 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
 ` ei2π/3 0 0 e-i2π/3`
 -1
 -1
 ` 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
 ` e-i2π/3 0 0 ei2π/3`
 -1
 -1
 ` 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
 ` 0 1 1 0`
 -i
 i
 ` 0 -1 1 0`
5
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 -i -i 0`
2120
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 -i
 i
 ` 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
 ` 0 ei2π/3 e-i2π/3 0`
 -i
 i
 ` 0 e-iπ/3 e-i2π/3 0`
7
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 ` 1 0 0 1`
 -1
 -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`
d3+001
 1
 1
 ` ei2π/3 0 0 e-i2π/3`
 1
 1
 ` 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`
d3-001
 1
 1
 ` e-i2π/3 0 0 ei2π/3`
 1
 1
 ` 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`
d2110
 1
 -1
 ` 0 1 1 0`
 i
 -i
 ` 0 1 -1 0`
11
 ` -1 1 0 0 1 0 0 0 -1`
 ` 0 i i 0`
d2120
 1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 i
 -i
 ` 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`
d2210
 1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 i
 -i
 ` 0 ei2π/3 eiπ/3 0`
k-Subgroupsmag
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