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The following tables indicate the number of symmetry operations of a given period for each point group. If two point groups possess the same elements in the following tables, then they are termed isomorphic. Sets of isomorphic groups are form an abstract group (eg C2h, C2v and D2).
The period of a symmetry operation is defined as the number of times that operation must be performed to be equivalent to the identity. In the following tables, each entry indicates the number of symmetry operations that has the indicated periodicity within the point group.
Symmetry Operation: |
E |
Cn |
Sn (n even) |
Sn (n odd) |
&sigma |
i |
Period: |
1 |
n |
n |
2n |
2 |
2 |
|
Period |
|
Period |
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
|
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
10 |
C1 |
1 |
|
|
|
|
|
|
C1h |
1 |
1 |
|
|
|
|
|
C2 |
1 |
1 |
|
|
|
|
|
C2h |
1 |
3 |
|
|
|
|
|
C3 |
1 |
|
2 |
|
|
|
|
C3h |
1 |
1 |
2 |
|
|
2 |
|
C4 |
1 |
1 |
|
2 |
|
|
|
C4h |
1 |
3 |
|
4 |
|
|
|
C5 |
1 |
|
|
|
4 |
|
|
C5h |
1 |
1 |
|
|
4 |
|
4 |
C6 |
1 |
1 |
2 |
|
|
2 |
|
C6h |
1 |
3 |
4 |
|
|
4 |
|
|
|
Period |
|
Period |
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
|
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
S2 |
1 |
1 |
|
|
|
|
|
C2v |
1 |
3 |
|
|
|
|
|
|
C3v |
1 |
3 |
2 |
|
|
|
S4 |
1 |
1 |
|
2 |
|
|
|
C4v |
1 |
5 |
|
2 |
|
|
|
|
C5v |
1 |
5 |
|
|
4 |
|
S6 |
1 |
1 |
2 |
|
|
2 |
|
C6v |
1 |
7 |
2 |
|
|
2 |
|
|
Period |
|
Period |
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
|
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
10 |
D2 |
1 |
3 |
|
|
|
|
|
D2h |
1 |
7 |
|
|
|
|
D3 |
1 |
3 |
2 |
|
|
|
|
D3h |
1 |
7 |
2 |
|
|
2 |
D4 |
1 |
5 |
|
2 |
|
|
|
D4h |
1 |
11 |
|
4 |
|
|
D5 |
1 |
5 |
|
|
4 |
|
|
D5h |
1 |
11 |
|
|
4 |
|
4 |
D6 |
1 |
7 |
2 |
|
|
2 |
|
D6h |
1 |
15 |
4 |
|
|
4 |
|
Period |
|
Period |
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
|
Point group |
1 |
2 |
3 |
4 |
5 |
6 |
8 |
10 |
12 |
T |
1 |
3 |
8 |
|
|
|
|
D2d |
1 |
5 |
|
2 |
|
|
|
|
|
Th |
1 |
7 |
8 |
|
|
8 |
|
D3d |
1 |
7 |
2 |
|
|
2 |
|
|
|
O |
1 |
9 |
8 |
6 |
|
|
|
D4d |
1 |
9 |
|
2 |
|
|
4 |
|
|
Td |
1 |
9 |
8 |
6 |
|
|
|
D5d |
1 |
11 |
|
|
4 |
|
|
4 |
|
Oh |
1 |
19 |
8 |
12 |
|
8 |
|
D6d |
1 |
13 |
2 |
2 |
|
2 |
|
|
4 |
|