## Construction of chiral 4-polytopes with alternating or symmetric automorphism group

##### J. Algebr. Comb. 42(1), 225-244 (2015)
DOI: 10.1007/s10801-014-0579-5

### Summary

In this paper we describe a construction for finite abstract chiral 4-polytopes with Schläfli type $\{3,3,k\}$ (with tetrahedral facets) and with an alternating or symmetric group as automorphism group. We use it to prove that for all but finitely many $n$, both $A_n$ and $S_n$ are the automorphism groups of such a polytope. We also show that the vertex-figures of the polytopes obtained from our construction are chiral.

51M20, 20B30

### Keywords/Phrases

abstract polytopes, chiral polytopes, symmetric groups, alternating groups