Bilbao Crystallographic Server Representations

## Irreducible representations of the Double Point Group 422 (No. 12)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 GM1 A1 GM1 1 1 1 1 1 1 1 GM3 B1 GM2 1 1 -1 1 -1 1 -1 GM2 A2 GM3 1 1 1 -1 -1 1 1 GM4 B2 GM4 1 1 -1 -1 1 1 -1 GM5 E GM5 2 -2 0 0 0 2 0 GM7 E2 GM6 2 0 -√2 0 0 -2 √2 GM6 E1 GM7 2 0 √2 0 0 -2 -√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: 2001, d2001 C3: 4+001, 4-001 C4: 2010, 2100, d2010, d2100 C5: 2110, 21-10, d2110, d21-10 C6: d1 C7: d4+001, d4-001

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)
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`
2
 ` -1 0 0 0 -1 0 0 0 1`
 ` -i 0 0 i`
2001
 1
 1
 1
 1
 ` -1 0 0 -1`
 ` -i 0 0 i`
 ` -i 0 0 i`
3
 ` 0 -1 0 1 0 0 0 0 1`
 ` (1-i)√2/2 0 0 (1+i)√2/2`
4+001
 1
 -1
 1
 -1
 ` 0 -1 1 0`
 ` ei3π/4 0 0 e-i3π/4`
 ` e-iπ/4 0 0 eiπ/4`
4
 ` 0 1 0 -1 0 0 0 0 1`
 ` (1+i)√2/2 0 0 (1-i)√2/2`
4-001
 1
 -1
 1
 -1
 ` 0 1 -1 0`
 ` e-i3π/4 0 0 ei3π/4`
 ` eiπ/4 0 0 e-iπ/4`
5
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 -1 1 0`
2010
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 e-iπ/4 e-i3π/4 0`
 ` 0 ei3π/4 eiπ/4 0`
6
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 -i -i 0`
2100
 1
 1
 -1
 -1
 ` 0 -1 -1 0`
 ` 0 eiπ/4 ei3π/4 0`
 ` 0 e-i3π/4 e-iπ/4 0`
7
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 -(1+i)√2/2 (1-i)√2/2 0`
2110
 1
 -1
 -1
 1
 ` 1 0 0 -1`
 ` 0 -1 1 0`
 ` 0 -1 1 0`
8
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 -(1-i)√2/2 (1+i)√2/2 0`
2110
 1
 -1
 -1
 1
 ` -1 0 0 1`
 ` 0 i i 0`
 ` 0 i i 0`
9
 ` 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`
10
 ` -1 0 0 0 -1 0 0 0 1`
 ` i 0 0 -i`
d2001
 1
 1
 1
 1
 ` -1 0 0 -1`
 ` i 0 0 -i`
 ` i 0 0 -i`
11
 ` 0 -1 0 1 0 0 0 0 1`
 ` -(1-i)√2/2 0 0 -(1+i)√2/2`
d4+001
 1
 -1
 1
 -1
 ` 0 -1 1 0`
 ` e-iπ/4 0 0 eiπ/4`
 ` ei3π/4 0 0 e-i3π/4`
12
 ` 0 1 0 -1 0 0 0 0 1`
 ` -(1+i)√2/2 0 0 -(1-i)√2/2`
d4-001
 1
 -1
 1
 -1
 ` 0 1 -1 0`
 ` eiπ/4 0 0 e-iπ/4`
 ` e-i3π/4 0 0 ei3π/4`
13
 ` -1 0 0 0 1 0 0 0 -1`
 ` 0 1 -1 0`
d2010
 1
 1
 -1
 -1
 ` 0 1 1 0`
 ` 0 ei3π/4 eiπ/4 0`
 ` 0 e-iπ/4 e-i3π/4 0`
14
 ` 1 0 0 0 -1 0 0 0 -1`
 ` 0 i i 0`
d2100
 1
 1
 -1
 -1
 ` 0 -1 -1 0`
 ` 0 e-i3π/4 e-iπ/4 0`
 ` 0 eiπ/4 ei3π/4 0`
15
 ` 0 1 0 1 0 0 0 0 -1`
 ` 0 (1+i)√2/2 -(1-i)√2/2 0`
d2110
 1
 -1
 -1
 1
 ` 1 0 0 -1`
 ` 0 1 -1 0`
 ` 0 1 -1 0`
16
 ` 0 -1 0 -1 0 0 0 0 -1`
 ` 0 (1-i)√2/2 -(1+i)√2/2 0`
d2110
 1
 -1
 -1
 1
 ` -1 0 0 1`
 ` 0 -i -i 0`
 ` 0 -i -i 0`
k-Subgroupsmag