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

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
GM3+
Eg
GM3+
2
-1
0
2
-1
0
2
-1
0
2
-1
0
GM3-
Eu
GM3-
2
-1
0
2
-1
0
-2
1
0
-2
1
0
GM4+
2Eg
GM4
1
-1
-i
-1
1
i
1
-1
-i
-1
1
i
GM5+
1Eg
GM5
1
-1
i
-1
1
-i
1
-1
i
-1
1
-i
GM4-
2Eu
GM6
1
-1
-i
-1
1
i
-1
1
i
1
-1
-i
GM5-
1Eu
GM7
1
-1
i
-1
1
-i
-1
1
-i
1
-1
i
GM6+
E1g
GM8
2
1
0
-2
-1
0
2
1
0
-2
-1
0
GM6-
E1u
GM9
2
1
0
-2
-1
0
-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: 21-10, 2120, 2210
C4d1
C5d3+001d3-001
C6d21-10d2120d2210
C7: -1
C8: -3+001, -3-001
C9: m1-10, m120, m210
C10d-1
C11d-3+001d-3-001
C12dm1-10dm120dm210

List of pairs of conjugated irreducible representations

(*GM4,*GM5)
(*GM6,*GM7)
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(0)
GM5(0)
GM6(0)
GM7(0)
GM8(-1)
GM9(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
2
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
-1
-1
-1
-1
(
e-iπ/3 0
0 eiπ/3
)
(
e-iπ/3 0
0 eiπ/3
)
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
-1
-1
-1
-1
(
eiπ/3 0
0 e-iπ/3
)
(
eiπ/3 0
0 e-iπ/3
)
4
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
-i
i
-i
i
(
0 -1
1 0
)
(
0 -1
1 0
)
5
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2120
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
-i
i
-i
i
(
0 eiπ/3
ei2π/3 0
)
(
0 eiπ/3
ei2π/3 0
)
6
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2210
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
-i
i
-i
i
(
0 e-iπ/3
e-i2π/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
7
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
1
1
-1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
8
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
-1
-1
1
1
(
e-iπ/3 0
0 eiπ/3
)
(
ei2π/3 0
0 e-i2π/3
)
9
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
-1
-1
1
1
(
eiπ/3 0
0 e-iπ/3
)
(
e-i2π/3 0
0 ei2π/3
)
10
(
0 1 0
1 0 0
0 0 1
)
(
0 -(3-i)/2
(3+i)/2 0
)
m110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
-i
i
i
-i
(
0 -1
1 0
)
(
0 1
-1 0
)
11
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m120
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
-i
i
i
-i
(
0 eiπ/3
ei2π/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
12
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m210
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
-i
i
i
-i
(
0 e-iπ/3
e-i2π/3 0
)
(
0 ei2π/3
eiπ/3 0
)
13
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
-1
-1
-1
-1
(
-1 0
0 -1
)
(
-1 0
0 -1
)
14
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
1
1
1
1
(
ei2π/3 0
0 e-i2π/3
)
(
ei2π/3 0
0 e-i2π/3
)
15
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
1
1
1
1
(
e-i2π/3 0
0 ei2π/3
)
(
e-i2π/3 0
0 ei2π/3
)
16
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2110
1
1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
i
-i
i
-i
(
0 1
-1 0
)
(
0 1
-1 0
)
17
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2120
1
1
-1
-1
(
0 e-i2π/3
ei2π/3 0
)
(
0 e-i2π/3
ei2π/3 0
)
i
-i
i
-i
(
0 e-i2π/3
e-iπ/3 0
)
(
0 e-i2π/3
e-iπ/3 0
)
18
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2210
1
1
-1
-1
(
0 ei2π/3
e-i2π/3 0
)
(
0 ei2π/3
e-i2π/3 0
)
i
-i
i
-i
(
0 ei2π/3
eiπ/3 0
)
(
0 ei2π/3
eiπ/3 0
)
19
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
-1
-1
1
1
(
-1 0
0 -1
)
(
1 0
0 1
)
20
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
-1
1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
1
1
-1
-1
(
ei2π/3 0
0 e-i2π/3
)
(
e-iπ/3 0
0 eiπ/3
)
21
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
-1
1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
1
1
-1
-1
(
e-i2π/3 0
0 ei2π/3
)
(
eiπ/3 0
0 e-iπ/3
)
22
(
0 1 0
1 0 0
0 0 1
)
(
0 (3-i)/2
-(3+i)/2 0
)
dm110
1
-1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
i
-i
-i
i
(
0 1
-1 0
)
(
0 -1
1 0
)
23
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm120
1
-1
-1
1
(
0 e-i2π/3
ei2π/3 0
)
(
0 eiπ/3
e-iπ/3 0
)
i
-i
-i
i
(
0 e-i2π/3
e-iπ/3 0
)
(
0 eiπ/3
ei2π/3 0
)
24
(
-1 0 0
-1 1 0
0 0 1
)
(
0 -(3+i)/2
(3-i)/2 0
)
dm210
1
-1
-1
1
(
0 ei2π/3
e-i2π/3 0
)
(
0 e-iπ/3
eiπ/3 0
)
i
-i
-i
i
(
0 ei2π/3
eiπ/3 0
)
(
0 e-iπ/3
e-i2π/3 0
)
k-Subgroupsmag
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