Give examples of polytopes $\Delta$ in $\mathbb{AR}^3$ such that

Question

With Sym $\Delta$ of the set $\Delta$ consisting of all isometries of $\mathbb{AR}^n$ that map $\Delta$ onto $\Delta$,

1. Sym $\Delta$ acts transitively on the set of vertices of $\Delta$ but is intransitive on the set of faces.

2. Sym $\Delta$ acts transitively on the set of faces of $\Delta$ but is intransitive on the set of vertices.

3. Sym $\Delta$ is transitive on the set of edges of $\Delta$ but is intransitive on the set of faces.

Any help or hints would be appreciated.

Note: This is questions from a book but is not hw.

Answer 1

For the first one, look up Archimedean solids. The duals of these will give your second class of examples. For the final class, you need a 3-connected planar graph that is edge- but not vertex-transitive, its dual will be edge- but not face transitive. Any graph that is edge- but not vertex-transitive must be bipartite. I'd start with a graph with the vertices and faces of the cube as its vertices.