We consider polyhedral realizations of oriented regular maps with or without self-intersections in E3 whose symmetry group is a subsgroup of small index in their. automorphism group. The four classical kepler-poinsot polyhedra are the only ones of index 1. There are exactly five of Index 2, all with icosahedral symmetry group [W2] as the Kepler-poinsot polyhedra.
In this paper we show that there are no such polyhedral realizations with octahedral (tetrahedral) symmetry group and index 2 or 3 (2,3,4,5), which is best possible in the octahedral case.
Riemann Surface Symmetry Group Automorphism Group Hide Symmetry Regular Polyhedron
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