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Irreducible representations of the Double Point Group 3m (No. 19)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
GM1
A1
GM1
1
1
1
1
1
1
GM2
A2
GM2
1
1
-1
1
1
-1
GM3
E
GM3
2
-1
0
2
-1
0
GM6
2E
GM4
1
-1
-i
-1
1
i
GM5
1E
GM5
1
-1
i
-1
1
-i
GM4
E1
GM6
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: m1-10, m120, m210
C4d1
C5d3+001d3-001
C6dm1-10dm120dm210

List of pairs of conjugated irreducible representations

(*GM4,*GM5)
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)
GM2(1)
GM3(1)
GM4(0)
GM5(0)
GM6(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
(
1 0
0 1
)
1
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
(
ei2π/3 0
0 e-i2π/3
)
-1
-1
(
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
(
e-i2π/3 0
0 ei2π/3
)
-1
-1
(
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
)
m110
1
-1
(
0 1
1 0
)
-i
i
(
0 -1
1 0
)
5
(
1 -1 0
0 -1 0
0 0 1
)
(
0 -i
-i 0
)
m120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
-i
i
(
0 eiπ/3
ei2π/3 0
)
6
(
-1 0 0
-1 1 0
0 0 1
)
(
0 (3+i)/2
-(3-i)/2 0
)
m210
1
-1
(
0 ei2π/3
e-i2π/3 0
)
-i
i
(
0 e-iπ/3
e-i2π/3 0
)
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
(
1 0
0 1
)
-1
-1
(
-1 0
0 -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 0
0 e-i2π/3
)
1
1
(
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
)
d3-001
1
1
(
e-i2π/3 0
0 ei2π/3
)
1
1
(
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
)
dm110
1
-1
(
0 1
1 0
)
i
-i
(
0 1
-1 0
)
11
(
1 -1 0
0 -1 0
0 0 1
)
(
0 i
i 0
)
dm120
1
-1
(
0 e-i2π/3
ei2π/3 0
)
i
-i
(
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
)
dm210
1
-1
(
0 ei2π/3
e-i2π/3 0
)
i
-i
(
0 ei2π/3
eiπ/3 0
)
k-Subgroupsmag
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