Symmetry Pages

Periodicity

Symmetry Pages

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

Revised: © University of Newcastle upon Tyne, UK