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Irreducible representations of the Point Group 23 (No. 28)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
GM1
A
GM1
1
1
1
1
GM2
1E
GM2
1
1
-(1+i3)/2
-(1-i3)/2
GM3
2E
GM3
1
1
-(1-i3)/2
-(1+i3)/2
GM4
T
GM4
3
-1
0
0
(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, 2010, 2100
C3: 3--11-1, 3-1-1-1, 3--1-11, 3-111
C4: 3+1-1-1, 3+-1-11, 3+-11-1, 3+111

List of pairs of conjugated irreducible representations

(*GM2,*GM3)
Matrices of the representations of the group

The number in parenthesis 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)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
(
1 0 0
0 1 0
0 0 1
)
2
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
1
(
1 0 0
0 -1 0
0 0 -1
)
3
(
-1 0 0
0 1 0
0 0 -1
)
2010
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
4
(
1 0 0
0 -1 0
0 0 -1
)
2100
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
5
(
0 0 1
1 0 0
0 1 0
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
6
(
0 0 1
-1 0 0
0 -1 0
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
7
(
0 0 -1
-1 0 0
0 1 0
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
8
(
0 0 -1
1 0 0
0 -1 0
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
9
(
0 1 0
0 0 1
1 0 0
)
3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
10
(
0 -1 0
0 0 1
-1 0 0
)
3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
11
(
0 1 0
0 0 -1
-1 0 0
)
3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
12
(
0 -1 0
0 0 -1
1 0 0
)
3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
k-Subgroupsmag
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