TITLE: Irreducible factors of primitive permutation group representation
QUESTION [6 upvotes]: Consider a primitive permutation group $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$.
Consider the linear representation of $\Gamma$ by permutation matrices on $\Bbb R^n$ in the obvious way. 
What can be said about the decomposition of this representation into irreducible factors?
I am especially interested in the multiplicities of the factors: is it known whether any two irreducible factors in such a decomposition are non-isomorphic?

REPLY [4 votes]: Peripherally related: In a paper I wrote in 1997 about bases for primitive permutation groups, it is noted that if $G$ is a (faithful) primitive permutation group of degree $n$, and a complex irreducible character $\chi$ of $G$ occurs with multiplicity $m$ in the associated permutation character of degree $n$, then there is a base for $G$ of size at most $\frac{\chi(1)}{m}.$
Recall that a base for the permutation group $G$ acting on $\Omega$ is a subset $\beta$ of $\Omega$ such that only the identity element of $G$ fixes every element of $\beta$. 
Hence we obtain $|G| \leq n(n-1) \ldots (n+1 - \frac{\chi(1)}{m}) < n^{\frac{\chi(1)}{m}}.$