Bilbao Crystallographic Server Representations

## Irreducible representations of the Point Group 6 (No. 21)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 GM1 A GM1 1 1 1 1 1 1 GM4 B GM2 1 -1 1 1 -1 -1 GM5 2E1 GM3 1 1 -(1+i√3)/2 -(1-i√3)/2 -(1-i√3)/2 -(1+i√3)/2 GM2 2E2 GM4 1 -1 -(1+i√3)/2 -(1-i√3)/2 (1-i√3)/2 (1+i√3)/2 GM6 1E1 GM5 1 1 -(1-i√3)/2 -(1+i√3)/2 -(1+i√3)/2 -(1-i√3)/2 GM3 1E2 GM6 1 -1 -(1-i√3)/2 -(1+i√3)/2 (1+i√3)/2 (1-i√3)/2
 (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 C3: 3-001 C4: 3+001 C5: 6-001 C6: 6+001

List of pairs of conjugated irreducible representations

(*GM3,*GM5)
(*GM4,*GM6)
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(0)
GM4(0)
GM5(0)
GM6(0)
1
 ` 1 0 0 0 1 0 0 0 1`
1
 1
 1
 1
 1
 1
 1
2
 ` 0 -1 0 1 -1 0 0 0 1`
3+001
 1
 1
 ei2π/3
 ei2π/3
 e-i2π/3
 e-i2π/3
3
 ` -1 1 0 -1 0 0 0 0 1`
3-001
 1
 1
 e-i2π/3
 e-i2π/3
 ei2π/3
 ei2π/3
4
 ` -1 0 0 0 -1 0 0 0 1`
2001
 1
 -1
 1
 -1
 1
 -1
5
 ` 0 1 0 -1 1 0 0 0 1`
6-001
 1
 -1
 ei2π/3
 e-iπ/3
 e-i2π/3
 eiπ/3
6
 ` 1 -1 0 1 0 0 0 0 1`
6+001
 1
 -1
 e-i2π/3
 eiπ/3
 ei2π/3
 e-iπ/3
k-Subgroupsmag