Representation of multiply transitive group 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?
 A: If the action is primitive, this means only that a point stabilizer is a maximal subgroup. I do not think that this puts much restriction on $\theta,$ but others may be able to say more than I can. In general, the character inner product $1 + \langle \theta, \theta \rangle$ is the rank of $G$ as a permutation group (when $G$ is transitive). This is the number of orbits that a point stabilizer $G_{\alpha}$ has on the points other than $\alpha.$ Higher transitivity puts further restrictions on how $\theta$ restricts to $G_{\alpha}$, but I at least do not know of many easily described properties in terms of $k$-transitivity for $k >2$ which can be attributed to $\theta$. 
A: While, as Geoff has said in his answer, the only nice properties of $\chi$ that come to mind if the action is $3$-transitive, say, are about what happens if $\chi$ is restricted to a point stabiliser, $k$-transitive sets are nice for a different reasons. Namely, you can produce more irreducible characters than just $\chi$ out of them. E.g. you can consider the set of unordered pairs of elements of $X$ (or more generally, the set of $l$-element subsets for $l\lt k$). Of course, the action is not quite transitive, because e.g. the pair $(x,x)$ can only be sent to another pair with equal entries, but you can analyse this defect and obtain explicit descriptions of other irreducible characters.
For a guided solution with nice results in the case of symmetric groups, see exercise 7 on one of my representation theory exercise sheets.
