We give conditions for oriented labeled graphs that must be satisfied in order that such are permutations representations for groups of automorphisms of chiral or orientably regular polytopes. We develop a technique for construction of highly symmetric polytopes using such graphs. In particular, we construct chiral polytopes with the automorphism group Sn for each n>5, an infinite family of finite chiral polytopes of rank 4, a polytope of rank 5, as well as several infinite chiral polyhedra.
Pellicer et al. (Wed,) studied this question.
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