Geodesic
Construction of chiral 4-polytopes with alternating or symmetric automorphism group
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 225-244.

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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.
Classification : 51M20, 20B30
Keywords: abstract polytopes, chiral polytopes, symmetric groups, alternating groups 
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Conder, Marston; Hubard, Isabel; O'Reilly-Regueiro, Eugenia; Pellicer, Daniel. Construction of chiral 4-polytopes with alternating or symmetric automorphism group. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 1, pp. 225-244. http://geodesic.mathdoc.fr/item/JAC_2015__42_1_a3/