Symmetry groups of maps

For finite closed maps on the sphere, there is a complete classification of possible symmetry groups. For finite closed maps on the torus, we can describe the possible symmetry groups of their universal covers. 

We remind that an automorphism of a simple graph is a permutation of the vertices preserving adjacencies between vertices. For plane graphs, we require also that faces are sent to faces but for 3-connected graphs this condition is redundant. Recall that Aut(G) denotes the group of automorphisms of G. 

The list of point groups is split into two classes seven infinite families and seven sporadic cases. Every point group contains a normal subgroup formed by its rotations. 

1 The group Cm is the cyclic group of rotations by angle -k with 0 k m — 1 around a fixed axis A. 

2 The group Cmh is generated by Cm and a reflection of plane P with P being orthogonal to A. 

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