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Irreducible representations of the Double Point Group 4mm (No. 13)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A1
GM1
1
1
1
1
1
1
1
GM3
B1
GM2
1
1
-1
1
-1
1
-1
GM4
B2
GM3
1
1
-1
-1
1
1
-1
GM2
A2
GM4
1
1
1
-1
-1
1
1
GM5
E
GM5
2
-2
0
0
0
2
0
GM7
E2
GM6
2
0
-2
0
0
-2
2
GM6
E1
GM7
2
0
2
0
0
-2
-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: 2001d2001
C3: 4+001, 4-001
C4: m010, m100dm010dm100
C5: m110, m1-10dm110dm1-10
C6d1
C7d4+001d4-001

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)
GM6(-1)
GM7(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
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
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
3
(
0 -1 0
1 0 0
0 0 1
)
(
(1-i)2/2 0
0 (1+i)2/2
)
4+001
1
-1
-1
1
(
0 -1
1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
4
(
0 1 0
-1 0 0
0 0 1
)
(
(1+i)2/2 0
0 (1-i)2/2
)
4-001
1
-1
-1
1
(
0 1
-1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m010
1
1
-1
-1
(
0 1
1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
6
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
7
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m110
1
-1
1
-1
(
1 0
0 -1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m110
1
-1
1
-1
(
-1 0
0 1
)
(
0 i
i 0
)
(
0 i
i 0
)
9
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
11
(
0 -1 0
1 0 0
0 0 1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
-1
-1
1
(
0 -1
1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
12
(
0 1 0
-1 0 0
0 0 1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
-1
-1
1
(
0 1
-1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
13
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm010
1
1
-1
-1
(
0 1
1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
14
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
15
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm110
1
-1
1
-1
(
1 0
0 -1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm110
1
-1
1
-1
(
-1 0
0 1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
k-Subgroupsmag
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