Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Double Point Group 32 (No. 18)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
GM1
A1
GM1
1
1
1
1
1
1
GM2
A2
GM2
1
1
-1
1
1
-1
GM3
E
GM3
2
-1
0
2
-1
0
GM6
2E
GM4
1
-1
-i
-1
1
i
GM5
1E
GM5
1
-1
i
-1
1
-i
GM4
E1
GM6
2
1
0
-2
-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: 3+001, 3-001
C3: 21-10, 2120, 2210
C4d1
C5d3+001d3-001
C6d21-10d2120d2210

List of pairs of conjugated irreducible representations

(*GM4,*GM5)
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(0)
GM5(0)
GM6(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
(
1 0
0 1
)
1
1
(
1 0
0 1
)
2
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
1
(
ei2π/3 0
0 e-i2π/3
)
-1
-1
(
e-iπ/3 0
0 eiπ/3
)
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
-1
-1
(
eiπ/3 0
0 e-iπ/3
)
4
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2110
1
-1
(
0 1
1 0
)
-i
i
(
0 -1
1 0
)
5
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
-i
i
(
0 eiπ/3
ei2π/3 0
)
6
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2210
1
-1
(
0 ei2π/3
e-i2π/3 0
)
-i
i
(
0 e-iπ/3
e-i2π/3 0
)
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
(
1 0
0 1
)
-1
-1
(
-1 0
0 -1
)
8
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
(
ei2π/3 0
0 e-i2π/3
)
1
1
(
ei2π/3 0
0 e-i2π/3
)
9
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
1
1
(
e-i2π/3 0
0 ei2π/3
)
10
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2110
1
-1
(
0 1
1 0
)
i
-i
(
0 1
-1 0
)
11
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
i
-i
(
0 e-i2π/3
e-iπ/3 0
)
12
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2210
1
-1
(
0 ei2π/3
e-i2π/3 0
)
i
-i
(
0 ei2π/3
eiπ/3 0
)
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
Licencia de Creative Commons
For comments, please mail to
administrador.bcs@ehu.eus