Take any undirected graph $G$. We say that $G$ is vertex-transitive iff for every vertices $v,w$ there is an automorphism on $G$ that maps $v$ to $w$. We say that $G$ is edge-transitive iff for every edge $e,f$ there is an automorphism on $G$ that maps $e$ to $f$. We say that $G$ is edge-flip-invariant iff for every edge with endpoints $v,w$ there is an automorphism on $G$ that maps $v$ to $w$ and maps $w$ to $v$.
Upon seeing these three kinds of symmetry, I had a curious question:
Question: What is the smallest $n$ such that there is a graph with $n$ vertices that is vertex-transitive but neither edge-transitive nor edge-flip-invariant?
It is clearly vertex-transitive, since every vertex is a vertex of a square. It is also not edge-transitive, since an edge between two triangles cannot be mapped by an automorphism to an edge next to a square. And it is not edge-flip-invariant, since no automorphism can flip an edge that is next to a triangle that is surrounded by triangles.
But is there a smaller graph with this property? I had found the snub cube by looking through 'nice' polyhedra (so that it is easy to verify vertex-transitivity), and I am unsure whether there is a better way of finding such graphs.