Generalized CPR-graphs and applications

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 $S_n$ 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.