Smallest graph that is vertex-transitive but neither edge-transitive nor edge-flip-invariant?

I think the following graph with $12$ vertices does the job, but I don't know if it is minimal.

It is basically a hexagonal (anti)prism with extra diagonals. Label the vertices $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ and $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$. The edges are $\{A_i, A_{i+1}\}$, $\{B_i, B_{i+1}\}$, $\{A_i, B_i\}$, $\{A_i, B_{i+1}\}$, $\{A_i, B_{i+3}\}$, where the indices are modulo $6$.

Here is a picture to be wrapped around a cylinder, connecting the left and right sides together.

enter image description here

I don't think this kind of construction can work using a prism with fewer sides without introducing a mirror symmetry which would make it edge-flip-invariant.

Tags:

Optimization

Graph Theory

Symmetry

Extremal Graph Theory

Graph Isomorphism

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