Every polyhedron of genus zero has a unique canonical form which can be achieved by moving its vertices in such a way that its faces remain flat, the vertices themselves remain distinct, all of its edges are tangent to a common sphere, and the centroid of the edge tangency points is the center of that sphere. Besides being unique, the canonical form also brings out the maximum symmetry in a polyhedron. This is because any symmetry planes or axes in a non-canonical polyhedron are preserved in its canonical form [1].

To produce the table below, all 3-connected planar simple graphs with up to 12 vertices were generated using the plantri software developed by Gunnar Brinkmann and Brendan McKay. From each graph, a canonical polyhedron was produced and its symmetry determined. In canonical form, a polyhedron and its dual have the same symmetry, so the table shows the counts of all canonical polyhedra with up to 12 faces, cross tabulated by symmetry type and number of faces.