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Irreducible representations of the Point Group 432 (No. 30)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
GM1
A1
GM1
1
1
1
1
1
GM2
A2
GM2
1
1
-1
1
-1
GM3
E
GM3
2
2
0
-1
0
GM4
T1
GM4
3
-1
-1
0
1
GM5
T2
GM5
3
-1
1
0
-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 H. T. Stokes, B. J. Campbell, and R. Cordes (2013) Acta Cryst. A. 69, 388-395 for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001, 2010, 2100
C3: 201-1, 2011, 21-10, 2-101, 2101, 2110
C4: 3--11-1, 3-1-1-1, 3+1-1-1, 3+-1-11, 3+-11-1, 3+111, 3--1-11, 3-111
C5: 4+001, 4-010, 4+010, 4-001, 4+100, 4-100

Matrices of the representations of the group

N
General position
Seitz Symbol
GM1(1)
GM2(1)
GM3(1)
GM4(1)
GM5(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
(
1 0
0 1
)
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
)
2
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
(
1 0
0 1
)
(
1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
3
(
-1 0 0
0 1 0
0 0 -1
)
2010
1
1
(
1 0
0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
4
(
1 0 0
0 -1 0
0 0 -1
)
2100
1
1
(
1 0
0 1
)
(
-1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
5
(
0 0 1
1 0 0
0 1 0
)
3+111
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
6
(
0 0 1
-1 0 0
0 -1 0
)
3+111
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
7
(
0 0 -1
-1 0 0
0 1 0
)
3+111
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
8
(
0 0 -1
1 0 0
0 -1 0
)
3+111
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
9
(
0 1 0
0 0 1
1 0 0
)
3-111
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
10
(
0 -1 0
0 0 1
-1 0 0
)
3-111
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
11
(
0 1 0
0 0 -1
-1 0 0
)
3-111
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
12
(
0 -1 0
0 0 -1
1 0 0
)
3-111
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
13
(
0 1 0
1 0 0
0 0 -1
)
2110
1
-1
(
0 1
1 0
)
(
-1 0 0
0 0 1
0 1 0
)
(
1 0 0
0 0 -1
0 -1 0
)
14
(
0 -1 0
-1 0 0
0 0 -1
)
2110
1
-1
(
0 1
1 0
)
(
-1 0 0
0 0 -1
0 -1 0
)
(
1 0 0
0 0 1
0 1 0
)
15
(
0 1 0
-1 0 0
0 0 1
)
4-001
1
-1
(
0 1
1 0
)
(
1 0 0
0 0 1
0 -1 0
)
(
-1 0 0
0 0 -1
0 1 0
)
16
(
0 -1 0
1 0 0
0 0 1
)
4+001
1
-1
(
0 1
1 0
)
(
1 0 0
0 0 -1
0 1 0
)
(
-1 0 0
0 0 1
0 -1 0
)
17
(
1 0 0
0 0 1
0 -1 0
)
4-100
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 -1
0 1 0
1 0 0
)
(
0 0 1
0 -1 0
-1 0 0
)
18
(
-1 0 0
0 0 1
0 1 0
)
2011
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 1
0 -1 0
1 0 0
)
(
0 0 -1
0 1 0
-1 0 0
)
19
(
-1 0 0
0 0 -1
0 -1 0
)
2011
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 -1
0 -1 0
-1 0 0
)
(
0 0 1
0 1 0
1 0 0
)
20
(
1 0 0
0 0 -1
0 1 0
)
4+100
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 0 1
0 1 0
-1 0 0
)
(
0 0 -1
0 -1 0
1 0 0
)
21
(
0 0 1
0 1 0
-1 0 0
)
4+010
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1 0
1 0 0
0 0 1
)
(
0 1 0
-1 0 0
0 0 -1
)
22
(
0 0 1
0 -1 0
1 0 0
)
2101
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 1 0
1 0 0
0 0 -1
)
(
0 -1 0
-1 0 0
0 0 1
)
23
(
0 0 -1
0 1 0
1 0 0
)
4-010
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 1 0
-1 0 0
0 0 1
)
(
0 -1 0
1 0 0
0 0 -1
)
24
(
0 0 -1
0 -1 0
-1 0 0
)
2101
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 1 0
1 0 0
0 0 1
)
k-Subgroupsmag
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