The 57 groups recognized in MOPAC are given in Table 1.
Table 1:
Point Groups available within Symmetry Code
C_{1} 
C_{s} 
C_{i} 




O 

C_{2} 
C_{2v} 
C_{2h} 
D_{2} 
D_{2d} 
D_{2h} 

T 

C_{3} 
C_{3v} 
C_{3h} 
D_{3} 
D_{3d} 
D_{3h} 

T_{d} 

C_{4} 
C_{4v} 
C_{4h} 
D_{4} 
D_{4d} 
D_{4h} 
S_{4} 
T_{h} 

C_{5} 
C_{5v} 
C_{5h} 
D_{5} 
D_{5d} 
D_{5h} 

O_{h} 

C_{6} 
C_{6v} 
C_{6h} 
D_{6} 
D_{6d} 
D_{6h} 
S_{6} 
I 
I_{h} 
C_{7} 
C_{7v} 
C_{7h} 
D_{7} 
D_{7d} 
D_{7h} 

C 
D 
C_{8} 
C_{8v} 
C_{8h} 
D_{8} 

D_{8h} 
S_{8} 
R_{3} 

Each point group is represented by a subset of the associated pointgroup table. For example, the group D_{2h} is represented by the subset shown in Table 2. The operations selected for the subgroup are the identity, E, and that minimum set of operations which is sufficient to allow all the operations to be generated as products of these operations. Thus, for the highest finite point group, I_{h}, the generating operations are: E, I, C_{3}, and C_{5}. Although it is not obvious, all 120 operations of the group can be generated as products of these four operations.
Each pointgroup is assumed to contain the totally symmetric representation, here A_{1g}. Operations are represented as 3x3 Euler matrices, thus C_{2x}, C_{2y} and C_{2z} would be represented as in the Figure All operations not given can be generated as products of operations already known, thus C_{2x} = C_{2y} C_{2z}.
Figure:
Representation of Symmetry OperationsC_{2x}:  
1  0  0 
0  1  0 
0  0  1 
C_{2y}:  
1  0  0 
0  1  0 
0  0  1 
C_{2z}:  
1  0  0 
0  1  0 
0  0  1 
In order to minimize storage, the characters in character tables are stored separately from the point groups. This allows, e.g., C_{2v}, C_{2h}, and D_{2} to use the same character table.