Bilbao Crystallographic Server Representations

## Irreducible representations of the Double Point Group 62m (No. 26)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 C9 GM1 A1' GM1 1 1 1 1 1 1 1 1 1 GM2 A2' GM2 1 1 -1 -1 1 -1 1 1 -1 GM4 A2'' GM3 1 1 -1 -1 -1 1 1 1 -1 GM3 A1'' GM4 1 1 1 1 -1 -1 1 1 1 GM6 E' GM5 2 -1 2 -1 0 0 2 -1 -1 GM5 E'' GM6 2 -1 -2 1 0 0 2 -1 1 GM9 E3 GM7 2 -2 0 0 0 0 -2 2 0 GM8 E2 GM8 2 1 0 -√3 0 0 -2 -1 √3 GM7 E1 GM9 2 1 0 √3 0 0 -2 -1 -√3
 (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, 3-001 C3: m001, dm001 C4: -6-001, -6+001 C5: 2110, 2100, 2010, d2110, d2100, d2010 C6: m1-10, m120, m210, dm1-10, dm120, dm210 C7: d1 C8: d3+001, d3-001 C9: d-6-001, d-6+001

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(1)
GM5(1)
GM6(1)
GM7(-1)
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 0 0 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 0 0 -1`
 ` e-iπ/3 0 0 eiπ/3`
 ` eiπ/3 0 0 e-iπ/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 0 0 -1`
 ` eiπ/3 0 0 e-iπ/3`
 ` e-iπ/3 0 0 eiπ/3`
4
 ` 1 0 0 0 1 0 0 0 -1`
 ` -i 0 0 i`
m001
 1
 -1
 -1
 1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` -i 0 0 i`
 ` -i 0 0 i`
 ` -i 0 0 i`
5
 ` 0 -1 0 1 -1 0 0 0 -1`
 ` (√3-i)/2 0 0 (√3+i)/2`
6-001
 1
 -1
 -1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 ` i 0 0 -i`
 ` e-i5π/6 0 0 ei5π/6`
 ` e-iπ/6 0 0 eiπ/6`
6
 ` -1 1 0 -1 0 0 0 0 -1`
 ` (√3+i)/2 0 0 (√3-i)/2`
6+001
 1
 -1
 -1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 ` -i 0 0 i`
 ` ei5π/6 0 0 e-i5π/6`
 ` eiπ/6 0 0 e-iπ/6`
7
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 -(1+i√3)/2 (1-i√3)/2 0`
2110
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
8
 ` 1 -1 0 0 -1 0 0 0 -1`
 ` 0 -1 1 0`
2100
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 1 -1 0`
 ` 0 e-i2π/3 e-iπ/3 0`
 ` 0 ei2π/3 eiπ/3 0`
9
 ` -1 0 0 -1 1 0 0 0 -1`
 ` 0 -(1-i√3)/2 (1+i√3)/2 0`
2010
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 -1 1 0`
 ` 0 e-iπ/3 e-i2π/3 0`
 ` 0 eiπ/3 ei2π/3 0`
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`
 ` 0 i i 0`
 ` 0 i i 0`
 ` 0 i i 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`
 ` 0 i i 0`
 ` 0 e-iπ/6 e-i5π/6 0`
 ` 0 e-i5π/6 e-iπ/6 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`
 ` 0 i i 0`
 ` 0 e-i5π/6 e-iπ/6 0`
 ` 0 e-iπ/6 e-i5π/6 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 0 0 -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 0 0 1`
 ` ei2π/3 0 0 e-i2π/3`
 ` e-i2π/3 0 0 ei2π/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 0 0 1`
 ` e-i2π/3 0 0 ei2π/3`
 ` ei2π/3 0 0 e-i2π/3`
16
 ` 1 0 0 0 1 0 0 0 -1`
 ` i 0 0 -i`
dm001
 1
 -1
 -1
 1
 ` 1 0 0 1`
 ` -1 0 0 -1`
 ` i 0 0 -i`
 ` i 0 0 -i`
 ` i 0 0 -i`
17
 ` 0 -1 0 1 -1 0 0 0 -1`
 ` -(√3-i)/2 0 0 -(√3+i)/2`
d6-001
 1
 -1
 -1
 1
 ` ei2π/3 0 0 e-i2π/3`
 ` e-iπ/3 0 0 eiπ/3`
 ` -i 0 0 i`
 ` eiπ/6 0 0 e-iπ/6`
 ` ei5π/6 0 0 e-i5π/6`
18
 ` -1 1 0 -1 0 0 0 0 -1`
 ` -(√3+i)/2 0 0 -(√3-i)/2`
d6+001
 1
 -1
 -1
 1
 ` e-i2π/3 0 0 ei2π/3`
 ` eiπ/3 0 0 e-iπ/3`
 ` i 0 0 -i`
 ` e-iπ/6 0 0 eiπ/6`
 ` e-i5π/6 0 0 ei5π/6`
19
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 (1+i√3)/2 -(1-i√3)/2 0`
d2110
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 1 1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
20
 ` 1 -1 0 0 -1 0 0 0 -1`
 ` 0 1 -1 0`
d2100
 1
 1
 -1
 -1
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 e-i2π/3 ei2π/3 0`
 ` 0 -1 1 0`
 ` 0 eiπ/3 ei2π/3 0`
 ` 0 e-iπ/3 e-i2π/3 0`
21
 ` -1 0 0 -1 1 0 0 0 -1`
 ` 0 (1-i√3)/2 -(1+i√3)/2 0`
d2010
 1
 1
 -1
 -1
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 ei2π/3 e-i2π/3 0`
 ` 0 1 -1 0`
 ` 0 ei2π/3 eiπ/3 0`
 ` 0 e-i2π/3 e-iπ/3 0`
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`
 ` 0 -i -i 0`
 ` 0 -i -i 0`
 ` 0 -i -i 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`
 ` 0 -i -i 0`
 ` 0 ei5π/6 eiπ/6 0`
 ` 0 eiπ/6 ei5π/6 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`
 ` 0 -i -i 0`
 ` 0 eiπ/6 ei5π/6 0`
 ` 0 ei5π/6 eiπ/6 0`
k-Subgroupsmag