Let $G$ be a finite group acting on a finite set $X$. If the action is primitive then the stabilizers are maximal subgroups of $G$ (converse also true). Is there any criteria to get maximal subgroups as stabilizers, with some restriction on the "action"?
For ex. the action of $S_3$ on $X=\{1,2,3\}$ is primitive, and so stabilizers are maximal subgroups of $G$, which are Sylow-2 subgroups of $S_3$. But how can we obtain the maximal subgroup $\langle (123)\rangle$?