Bilbao Crystallographic Server arrow Representations


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

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A
GM1
1
1
1
1
1
1
1
GM2
1E
GM2
1
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
GM3
2E
GM3
1
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
GM4
T
GM4
3
-1
0
0
3
0
0
GM5
E
GM5
2
0
1
1
-2
-1
-1
GM7
2F
GM6
2
0
-(1-i3)/2
-(1+i3)/2
-2
(1-i3)/2
(1+i3)/2
GM6
1F
GM7
2
0
-(1+i3)/2
-(1-i3)/2
-2
(1+i3)/2
(1-i3)/2
(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, 2100d2001d2010d2100
C3: 3+111, 3+-11-1, 3+1-1-1, 3+-1-11
C4: 3-111, 3-1-1-1, 3--1-11, 3--11-1
C5d1
C6d3+111d3+-11-1d3+1-1-1d3+-1-11
C7d3-111d3-1-1-1d3--1-11d3--11-1

List of pairs of conjugated irreducible representations

(*GM2,*GM3)
(*GM6,*GM7)
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(0)
GM3(0)
GM4(1)
GM5(-1)
GM6(0)
GM7(0)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
(
1 0 0
0 1 0
0 0 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 0 0
0 -1 0
0 0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
3
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2010
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
4
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2100
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
5
(
0 0 1
1 0 0
0 1 0
)
(
(1-i)/2 -(1+i)/2
(1-i)/2 (1+i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
(
e-iπ/42/2 e-i3π/42/2
e-iπ/42/2 eiπ/42/2
)
(
ei5π/122/2 e-iπ/122/2
ei5π/122/2 ei11π/122/2
)
(
e-i11π/122/2 ei7π/122/2
e-i11π/122/2 e-i5π/122/2
)
6
(
0 0 1
-1 0 0
0 -1 0
)
(
(1+i)/2 -(1-i)/2
(1+i)/2 (1-i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
(
eiπ/42/2 ei3π/42/2
eiπ/42/2 e-iπ/42/2
)
(
ei11π/122/2 e-i7π/122/2
ei11π/122/2 ei5π/122/2
)
(
e-i5π/122/2 eiπ/122/2
e-i5π/122/2 e-i11π/122/2
)
7
(
0 0 -1
-1 0 0
0 1 0
)
(
(1+i)/2 (1-i)/2
-(1+i)/2 (1-i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
(
eiπ/42/2 e-iπ/42/2
e-i3π/42/2 e-iπ/42/2
)
(
ei11π/122/2 ei5π/122/2
e-iπ/122/2 ei5π/122/2
)
(
e-i5π/122/2 e-i11π/122/2
ei7π/122/2 e-i11π/122/2
)
8
(
0 0 -1
1 0 0
0 -1 0
)
(
(1-i)/2 (1+i)/2
-(1-i)/2 (1+i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
(
e-iπ/42/2 eiπ/42/2
ei3π/42/2 eiπ/42/2
)
(
ei5π/122/2 ei11π/122/2
e-i7π/122/2 ei11π/122/2
)
(
e-i11π/122/2 e-i5π/122/2
eiπ/122/2 e-i5π/122/2
)
9
(
0 1 0
0 0 1
1 0 0
)
(
(1+i)/2 (1+i)/2
-(1-i)/2 (1-i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
(
eiπ/42/2 eiπ/42/2
ei3π/42/2 e-iπ/42/2
)
(
e-i5π/122/2 e-i5π/122/2
eiπ/122/2 e-i11π/122/2
)
(
ei11π/122/2 ei11π/122/2
e-i7π/122/2 ei5π/122/2
)
10
(
0 -1 0
0 0 1
-1 0 0
)
(
(1-i)/2 -(1-i)/2
(1+i)/2 (1+i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
(
e-iπ/42/2 ei3π/42/2
eiπ/42/2 eiπ/42/2
)
(
e-i11π/122/2 eiπ/122/2
e-i5π/122/2 e-i5π/122/2
)
(
ei5π/122/2 e-i7π/122/2
ei11π/122/2 ei11π/122/2
)
11
(
0 1 0
0 0 -1
-1 0 0
)
(
(1+i)/2 -(1+i)/2
(1-i)/2 (1-i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
(
eiπ/42/2 e-i3π/42/2
e-iπ/42/2 e-iπ/42/2
)
(
e-i5π/122/2 ei7π/122/2
e-i11π/122/2 e-i11π/122/2
)
(
ei11π/122/2 e-iπ/122/2
ei5π/122/2 ei5π/122/2
)
12
(
0 -1 0
0 0 -1
1 0 0
)
(
(1-i)/2 (1-i)/2
-(1+i)/2 (1+i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
(
e-iπ/42/2 e-iπ/42/2
e-i3π/42/2 eiπ/42/2
)
(
e-i11π/122/2 e-i11π/122/2
ei7π/122/2 e-i5π/122/2
)
(
ei5π/122/2 ei5π/122/2
e-iπ/122/2 ei11π/122/2
)
13
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
14
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
(
1 0 0
0 -1 0
0 0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
15
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
17
(
0 0 1
1 0 0
0 1 0
)
(
-(1-i)/2 (1+i)/2
-(1-i)/2 -(1+i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
(
ei3π/42/2 eiπ/42/2
ei3π/42/2 e-i3π/42/2
)
(
e-i7π/122/2 ei11π/122/2
e-i7π/122/2 e-iπ/122/2
)
(
eiπ/122/2 e-i5π/122/2
eiπ/122/2 ei7π/122/2
)
18
(
0 0 1
-1 0 0
0 -1 0
)
(
-(1+i)/2 (1-i)/2
-(1+i)/2 -(1-i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
(
e-i3π/42/2 e-iπ/42/2
e-i3π/42/2 ei3π/42/2
)
(
e-iπ/122/2 ei5π/122/2
e-iπ/122/2 e-i7π/122/2
)
(
ei7π/122/2 e-i11π/122/2
ei7π/122/2 eiπ/122/2
)
19
(
0 0 -1
-1 0 0
0 1 0
)
(
-(1+i)/2 -(1-i)/2
(1+i)/2 -(1-i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
(
e-i3π/42/2 ei3π/42/2
eiπ/42/2 ei3π/42/2
)
(
e-iπ/122/2 e-i7π/122/2
ei11π/122/2 e-i7π/122/2
)
(
ei7π/122/2 eiπ/122/2
e-i5π/122/2 eiπ/122/2
)
20
(
0 0 -1
1 0 0
0 -1 0
)
(
-(1-i)/2 -(1+i)/2
(1-i)/2 -(1+i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
(
ei3π/42/2 e-i3π/42/2
e-iπ/42/2 e-i3π/42/2
)
(
e-i7π/122/2 e-iπ/122/2
ei5π/122/2 e-iπ/122/2
)
(
eiπ/122/2 ei7π/122/2
e-i11π/122/2 ei7π/122/2
)
21
(
0 1 0
0 0 1
1 0 0
)
(
-(1+i)/2 -(1+i)/2
(1-i)/2 -(1-i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
(
e-i3π/42/2 e-i3π/42/2
e-iπ/42/2 ei3π/42/2
)
(
ei7π/122/2 ei7π/122/2
e-i11π/122/2 eiπ/122/2
)
(
e-iπ/122/2 e-iπ/122/2
ei5π/122/2 e-i7π/122/2
)
22
(
0 -1 0
0 0 1
-1 0 0
)
(
-(1-i)/2 (1-i)/2
-(1+i)/2 -(1+i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
(
ei3π/42/2 e-iπ/42/2
e-i3π/42/2 e-i3π/42/2
)
(
eiπ/122/2 e-i11π/122/2
ei7π/122/2 ei7π/122/2
)
(
e-i7π/122/2 ei5π/122/2
e-iπ/122/2 e-iπ/122/2
)
23
(
0 1 0
0 0 -1
-1 0 0
)
(
-(1+i)/2 (1+i)/2
-(1-i)/2 -(1-i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
(
e-i3π/42/2 eiπ/42/2
ei3π/42/2 ei3π/42/2
)
(
ei7π/122/2 e-i5π/122/2
eiπ/122/2 eiπ/122/2
)
(
e-iπ/122/2 ei11π/122/2
e-i7π/122/2 e-i7π/122/2
)
24
(
0 -1 0
0 0 -1
1 0 0
)
(
-(1-i)/2 -(1-i)/2
(1+i)/2 -(1+i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
(
ei3π/42/2 ei3π/42/2
eiπ/42/2 e-i3π/42/2
)
(
eiπ/122/2 eiπ/122/2
e-i5π/122/2 ei7π/122/2
)
(
e-i7π/122/2 e-i7π/122/2
ei11π/122/2 e-iπ/122/2
)
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
Licencia de Creative Commons
For comments, please mail to
administrador.bcs@ehu.eus