Let $G$ be a finite group acting on a finite set $X$, and $\chi$ be the corresponding permutation character over the field $\mathbb{C}$.
1)If $G$ acts transitively, then $\chi=\mathbb{1}+\theta$, ($\mathbb{1}$ is trivial character of $G$); moreover the trivial character appears only once in $\chi$.
2)If $G$ acts doubly transitively, then the character $\theta$ is irreducible.
(Ref. Linear Representations of finite groups -Serre)
Question: What can be said about $\chi$ when the action is primitive, $3$-transitive, or in general $n$-trnasitive?