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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 C. J. Bradley, A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) 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

The number in parenthesis 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)
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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