Bilbao Crystallographic Server Representations

## Irreducible representations of the Point Group 6 (No. 22)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 GM1 A' GM1 1 1 1 1 1 1 GM4 A'' GM2 1 -1 1 -1 1 -1 GM2 2E' GM3 1 -(1+i√3)/2 -(1+i√3)/2 -(1-i√3)/2 -(1-i√3)/2 1 GM5 2E'' GM4 1 (1+i√3)/2 -(1+i√3)/2 (1-i√3)/2 -(1-i√3)/2 -1 GM3 1E' GM5 1 -(1-i√3)/2 -(1-i√3)/2 -(1+i√3)/2 -(1+i√3)/2 1 GM6 1E'' GM6 1 (1-i√3)/2 -(1-i√3)/2 (1+i√3)/2 -(1+i√3)/2 -1
 (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: -6+001 C3: 3-001 C4: -6-001 C5: 3+001 C6: m001

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`
m001
 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