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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 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: 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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