I have found that the subgroup of $S_3$ generated by a 3-cycle is $\{e,(123),(132)\}$ where $e$ is the identity but I can't find any graphs that have this group as their automorphism group.
I am a little bit confused as one of the automorphisms is $(123)$, the graph should have $3$ vertices but every graph I have tried to draw with $3$ vertices has $(123)$ as one of their automorphisms, have also have different automorphisms such as $(1)(23)$ so they do not satisfy the requirement in the title.