7
Lecture 2 1. Point Groups 2. Representations and matrices 3. Characters
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Lecture 21. Point Groups
2. Representations and matrices
3. Characters
Th
T
CCl4Td
SF6Oh Cube
Cubic (octahedral) symmetry
a
1
0
0
0
1
0
0
0
1
b
1
4
7
2
5
8
3
6
9
a b
1
4
7
2
5
8
3
6
9
a
1
0
0
0
1
0
0
0
1
b
1
4
7
2
5
8
3
6
9
a b
1
4
7
2
5
8
3
6
9
a
1
0
0
0
1
0
0
0
1
b
1
4
7
2
5
8
3
6
9
a b
1
4
7
2
5
8
3
6
9
Multiplication of matrices
1
0
0
0
1
0
0
0
1
E =
1
0
0
0
1
0
0
0
1
yz=
1
0
0
0
1
0
0
0
1
xz=
1
0
0
0
1
0
0
0
1
C2=
1
0
0
0
1
0
0
0
1
i =
a
1
0
0
0
1
0
0
0
1
b
1
0
0
0
1
0
0
0
1
a b
1
0
0
0
1
0
0
0
1
a
1
0
0
0
1
0
0
0
1
b
1
0
0
0
1
0
0
0
1
a b
1
0
0
0
1
0
0
0
1
a
1
0
0
0
1
0
0
0
1
b
1
0
0
0
1
0
0
0
1
a b
1
0
0
0
1
0
0
0
1
a
1
0
0
0
1
0
0
0
1
b
1
0
0
0
1
0
0
0
1
a b
1
0
0
0
1
0
0
0
1
yz× C2 = xz
yz×xz =C2
c
cos theta( )
sin theta( )
0
sin theta( )
cos theta( )
0
0
0
1
General Case of rotation
theta23
c d
12
12
3
0
1
23
12
0
0
0
1
c c
12
1
23
0
12
3
12
0
0
0
1
120°
