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Irreducible representations of the Point Group 43m (No. 31)

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
0
-1
GM5
T2
GM4
3
-1
-1
1
0
GM4
T1
GM5
3
-1
1
-1
0
(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001, 2010, 2100
C3: -4-100, -4+100, -4-001, -4+010, -4-010, -4+001
C4: m110, m101, m-101, m1-10, m011, m01-1
C5: 3--11-1, 3-1-1-1, 3+1-1-1, 3+-1-11, 3+-11-1, 3+111, 3--1-11, 3-111

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)
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
)
m110
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
)
m110
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
)
m011
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
)
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
)
19
(
-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
)
20
(
1 0 0
0 0 -1
0 -1 0
)
m011
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
)
m101
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
)
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
)
23
(
0 0 -1
0 1 0
-1 0 0
)
m101
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
)
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
)
k-Subgroupsmag
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