Let $G=\operatorname{Sym}(3)$ acting on $S=\{1,2,3\}$ and let $S'=\{(x,y): x,y\in S, x\neq y\}$. Now $G$ acts on $S'$ by taking $(x,y)^\rho = (x^\rho, y^\rho)$ for all $\rho \in G$.
Is this action primitive?
First we find the stabilizer of a point and try to show that it is not maximal in $G$. For example, $\operatorname{Stab}_G(1,2)$ is trivial because $(12), (13), (23)$ do not fix it, neither do $(123), (132)$ and in $S_3$ we only have those cycles. So $\operatorname{Stab}_G(1,2) < \operatorname{Alt}(3) < G$, so the action is not primitive. Is this correct?