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Irreducible representations of the Point Group 3m (No. 20)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
GM1+
A1g
GM1+
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
-1
-1
-1
GM2+
A2g
GM2+
1
1
-1
1
1
-1
GM2-
A2u
GM2-
1
1
-1
-1
-1
1
GM3+
Eg
GM3+
2
-1
0
2
-1
0
GM3-
Eu
GM3-
2
-1
0
-2
1
0
(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: 3-001, 3+001
C3: 2120, 21-10, 2210
C4: -1
C5: -3-001, -3+001
C6: m120, m1-10, m210

Matrices of the representations of the group

N
General position
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
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
(
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
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
4
(
0 -1 0
-1 0 0
0 0 -1
)
2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
5
(
-1 1 0
0 1 0
0 0 -1
)
2120
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
6
(
1 0 0
1 -1 0
0 0 -1
)
2210
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
7
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
8
(
0 1 0
-1 1 0
0 0 -1
)
3+001
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
9
(
1 -1 0
1 0 0
0 0 -1
)
3-001
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
10
(
0 1 0
1 0 0
0 0 1
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
11
(
1 -1 0
0 -1 0
0 0 1
)
m120
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
12
(
-1 0 0
-1 1 0
0 0 1
)
m210
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
k-Subgroupsmag
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