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Irreducible representations of the Double Point Group mmm (No. 8)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
1
-1
-1
-1
-1
-1
GM3+
B1g
GM2+
1
1
-1
-1
1
1
1
-1
-1
1
GM3-
B1u
GM2-
1
1
-1
-1
1
-1
-1
1
1
-1
GM4+
B3g
GM3+
1
-1
-1
1
1
1
-1
-1
1
1
GM4-
B3u
GM3-
1
-1
-1
1
1
-1
1
1
-1
-1
GM2+
B2g
GM4+
1
-1
1
-1
1
1
-1
1
-1
1
GM2-
B2u
GM4-
1
-1
1
-1
1
-1
1
-1
1
-1
GM5+
Eg
GM5
2
0
0
0
-2
2
0
0
0
-2
GM5-
Eu
GM6
2
0
0
0
-2
-2
0
0
0
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: 2010d2010
C4: 2100d2100
C5d1
C6: -1
C7: m001dm001
C8: m010dm010
C9: m100dm100
C10d-1

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)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5(-1)
GM6(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
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
-1
-1
-1
(
0 -1
1 0
)
(
0 -1
1 0
)
3
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2010
1
1
-1
-1
-1
-1
1
1
(
0 -i
-i 0
)
(
0 -i
-i 0
)
4
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2100
1
1
-1
-1
1
1
-1
-1
(
-i 0
0 i
)
(
-i 0
0 i
)
5
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
6
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
-1
1
-1
1
(
0 -1
1 0
)
(
0 1
-1 0
)
7
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m010
1
-1
-1
1
-1
1
1
-1
(
0 -i
-i 0
)
(
0 i
i 0
)
8
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m100
1
-1
-1
1
1
-1
-1
1
(
-i 0
0 i
)
(
i 0
0 -i
)
9
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
1
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
-1
-1
-1
(
0 1
-1 0
)
(
0 1
-1 0
)
11
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
-1
-1
-1
-1
1
1
(
0 i
i 0
)
(
0 i
i 0
)
12
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
-1
-1
1
1
-1
-1
(
i 0
0 -i
)
(
i 0
0 -i
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
14
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
-1
1
-1
1
(
0 1
-1 0
)
(
0 -1
1 0
)
15
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm010
1
-1
-1
1
-1
1
1
-1
(
0 i
i 0
)
(
0 -i
-i 0
)
16
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm100
1
-1
-1
1
1
-1
-1
1
(
i 0
0 -i
)
(
-i 0
0 i
)
k-Subgroupsmag
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