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Irreducible representations of the Double Point Group 3 (No. 17)

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+
2Eg
GM2+
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
GM2-
2Eu
GM2-
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
-1
(1-i3)/2
(1+i3)/2
-1
(1-i3)/2
(1+i3)/2
GM3+
1Eg
GM3+
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
GM3-
1Eu
GM3-
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
-1
(1+i3)/2
(1-i3)/2
-1
(1+i3)/2
(1-i3)/2
GM6+
Eg
GM4
1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
GM4+
1Eg
GM5
1
(1-i3)/2
(1+i3)/2
-1
-(1-i3)/2
-(1+i3)/2
1
(1-i3)/2
(1+i3)/2
-1
-(1-i3)/2
-(1+i3)/2
GM5+
2Eg
GM6
1
(1+i3)/2
(1-i3)/2
-1
-(1+i3)/2
-(1-i3)/2
1
(1+i3)/2
(1-i3)/2
-1
-(1+i3)/2
-(1-i3)/2
GM6-
Eu
GM7
1
-1
-1
-1
1
1
-1
1
1
1
-1
-1
GM4-
1Eu
GM8
1
(1-i3)/2
(1+i3)/2
-1
-(1-i3)/2
-(1+i3)/2
-1
-(1-i3)/2
-(1+i3)/2
1
(1-i3)/2
(1+i3)/2
GM5-
2Eu
GM9
1
(1+i3)/2
(1-i3)/2
-1
-(1+i3)/2
-(1-i3)/2
-1
-(1+i3)/2
-(1-i3)/2
1
(1+i3)/2
(1-i3)/2
(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
C3: 3-001
C4d1
C5d3+001
C6d3-001
C7: -1
C8: -3+001
C9: -3-001
C10d-1
C11d-3+001
C12d-3-001

List of pairs of conjugated irreducible representations

(*GM2+,*GM3+)
(*GM2-,*GM3-)
(*GM5,*GM6)
(*GM8,*GM9)
Matrices of the representations of the group

The number in parentheses 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+(0)
GM2-(0)
GM3+(0)
GM3-(0)
GM4(1)
GM5(0)
GM6(0)
GM7(1)
GM8(0)
GM9(0)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
1
1
1
1
1
2
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
-1
e-iπ/3
eiπ/3
-1
e-iπ/3
eiπ/3
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
-1
eiπ/3
e-iπ/3
-1
eiπ/3
e-iπ/3
4
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-1
5
(
0 1 0
-1 1 0
0 0 -1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
-1
e-iπ/3
eiπ/3
1
ei2π/3
e-i2π/3
6
(
1 -1 0
1 0 0
0 0 -1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
-1
eiπ/3
e-iπ/3
1
e-i2π/3
ei2π/3
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
8
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
1
ei2π/3
e-i2π/3
1
ei2π/3
e-i2π/3
9
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
1
e-i2π/3
ei2π/3
1
e-i2π/3
ei2π/3
10
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
1
11
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
1
ei2π/3
e-i2π/3
-1
e-iπ/3
eiπ/3
12
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
1
e-i2π/3
ei2π/3
-1
eiπ/3
e-iπ/3
k-Subgroupsmag
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