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Irreducible representations of the Point Group 6/mmm (No. 27)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
GM2+
A2g
GM2+
1
-1
1
1
-1
1
1
-1
1
1
-1
1
GM2-
A2u
GM2-
1
-1
1
1
-1
1
-1
1
-1
-1
1
-1
GM4+
B2g
GM3+
1
1
-1
1
-1
-1
1
1
-1
1
-1
-1
GM4-
B2u
GM3-
1
1
-1
1
-1
-1
-1
-1
1
-1
1
1
GM3+
B1g
GM4+
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
GM3-
B1u
GM4-
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
GM6+
E2g
GM5+
2
0
2
-1
0
-1
2
0
2
-1
0
-1
GM6-
E2u
GM5-
2
0
2
-1
0
-1
-2
0
-2
1
0
1
GM5+
E1g
GM6+
2
0
-2
-1
0
1
2
0
-2
-1
0
1
GM5-
E1u
GM6-
2
0
-2
-1
0
1
-2
0
2
1
0
-1
(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 H. T. Stokes, B. J. Campbell, and R. Cordes (2013) Acta Cryst. A. 69, 388-395 for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2010, 2110, 2100
C3: 2001
C4: 3-001, 3+001
C5: 2120, 21-10, 2210
C6: 6-001, 6+001
C7: -1
C8: m010, m110, m100
C9: m001
C10: -3-001, -3+001
C11: m120, m1-10, m210
C12: -6-001, -6+001

Matrices of the representations of the group

N
General position
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5+(1)
GM5-(1)
GM6+(1)
GM6-(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
2
(
0 -1 0
1 -1 0
0 0 1
)
3+001
1
1
1
1
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
3
(
-1 1 0
-1 0 0
0 0 1
)
3-001
1
1
1
1
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
4
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
1
1
-1
-1
-1
-1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
5
(
0 1 0
-1 1 0
0 0 1
)
6-001
1
1
1
1
-1
-1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
6
(
1 -1 0
1 0 0
0 0 1
)
6+001
1
1
1
1
-1
-1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
7
(
0 1 0
1 0 0
0 0 -1
)
2110
1
1
-1
-1
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
1 0
)
8
(
1 -1 0
0 -1 0
0 0 -1
)
2100
1
1
-1
-1
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
9
(
-1 0 0
-1 1 0
0 0 -1
)
2010
1
1
-1
-1
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
10
(
0 -1 0
-1 0 0
0 0 -1
)
2110
1
1
-1
-1
-1
-1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
11
(
-1 1 0
0 1 0
0 0 -1
)
2120
1
1
-1
-1
-1
-1
1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
12
(
1 0 0
1 -1 0
0 0 -1
)
2210
1
1
-1
-1
-1
-1
1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
14
(
0 1 0
-1 1 0
0 0 -1
)
3+001
1
-1
1
-1
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
15
(
1 -1 0
1 0 0
0 0 -1
)
3-001
1
-1
1
-1
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
16
(
1 0 0
0 1 0
0 0 -1
)
m001
1
-1
1
-1
-1
1
-1
1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
17
(
0 -1 0
1 -1 0
0 0 -1
)
6-001
1
-1
1
-1
-1
1
-1
1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
18
(
-1 1 0
-1 0 0
0 0 -1
)
6+001
1
-1
1
-1
-1
1
-1
1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
19
(
0 -1 0
-1 0 0
0 0 1
)
m110
1
-1
-1
1
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 -1
-1 0
)
20
(
-1 1 0
0 1 0
0 0 1
)
m100
1
-1
-1
1
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
21
(
1 0 0
1 -1 0
0 0 1
)
m010
1
-1
-1
1
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
22
(
0 1 0
1 0 0
0 0 1
)
m110
1
-1
-1
1
-1
1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 1
1 0
)
23
(
1 -1 0
0 -1 0
0 0 1
)
m120
1
-1
-1
1
-1
1
1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
24
(
-1 0 0
-1 1 0
0 0 1
)
m210
1
-1
-1
1
-1
1
1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
k-Subgroupsmag
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