# Are there half-transitive convex polytopes?

I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $$P\subseteq\Bbb R^d$$ is the graph consisting of the polytope's vertices, two are adjacent if they lie on a common edge in $$P$$.

The orthogonal symmetry group of $$P$$ induces a permutation symmetry group on its graph. We say that a polytope is vertex-, edge-, arc- and/or half-transitive if the symmetry group induced on its graph acts in the respective way.

Question: Are there half-transitive polytopes, i.e. polytopes which are vertex- and edge-transitive, but not arc-transitive?

I looked a bit into chiral polytopes (i.e. two flag orbits), but I read somewhere that these only exist for abstract polytopes, and not for convex ones. However, I think that chirality is a much stronger requirement for a polytope than half-transitivity (as the latter only speaks about vertices and edges instead of flags).

• I don't see why the graph needs to be half-transitive. The symmetry group of the polytope might not induce the full group of the graph. Graphs with a half-transitive subgroup are much more common. (Cycles, for example.) – verret Jan 4 at 18:23
• @verret You are absolutely right. I edited that part out. – M. Winter Jan 4 at 18:38