Bilbao Crystallographic Server Representations

## Irreducible representations of the Double Point Group 3 (No. 16)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 GM1 A1 GM1 1 1 1 1 1 1 GM2 2E GM2 1 -(1-i√3)/2 -(1+i√3)/2 1 -(1-i√3)/2 -(1+i√3)/2 GM3 1E GM3 1 -(1+i√3)/2 -(1-i√3)/2 1 -(1+i√3)/2 -(1-i√3)/2 GM6 E GM4 1 -1 -1 -1 1 1 GM4 1E GM5 1 (1-i√3)/2 (1+i√3)/2 -1 -(1-i√3)/2 -(1+i√3)/2 GM5 2E GM6 1 (1+i√3)/2 (1-i√3)/2 -1 -(1+i√3)/2 -(1-i√3)/2
 (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: 3+001 C3: 3-001 C4: d1 C5: d3+001 C6: d3-001

List of pairs of conjugated irreducible representations

(*GM2,*GM3)
(*GM5,*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(0)
GM3(0)
GM4(1)
GM5(0)
GM6(0)
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 1
 1
 1
 1
2
 ` 0 -1 0 1 -1 0 0 0 1`
 ` (1+i√3)/2 0 0 (1-i√3)/2`
3+001
 1
 ei2π/3
 e-i2π/3
 -1
 e-iπ/3
 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
 e-i2π/3
 ei2π/3
 -1
 eiπ/3
 e-iπ/3
4
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 1
 -1
 -1
 -1
5
 ` 0 -1 0 1 -1 0 0 0 1`
 ` -(1+i√3)/2 0 0 -(1-i√3)/2`
d3+001
 1
 ei2π/3
 e-i2π/3
 1
 ei2π/3
 e-i2π/3
6
 ` -1 1 0 -1 0 0 0 0 1`
 ` -(1-i√3)/2 0 0 -(1+i√3)/2`
d3-001
 1
 e-i2π/3
 ei2π/3
 1
 e-i2π/3
 ei2π/3
k-Subgroupsmag