Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5⩽n⩽9 are available. Two observations arise when we look at the results: (1) for n⩾5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n⩾7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n⩾7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3⩽r⩽n−1.