It is well known that a finite 4-times transitive permutation group is Matthieu, symmetric, or alternating. Another way of stating this is that the set $$ \Omega = \{(x_1, x_2, x_3, x_4), 1\leq x_i\leq n, \mbox{$x_1, \ldots, x_4$ pairwise different}\} $$ has the property that $S_n$ acts transitively on $\Omega$, and every subgroup $U<S_n$, which also acts transitively on $\Omega$ is either $A_n$ or $S_n$, or one of a finite list of exceptions.

My question is: For which other simple groups does a similar statement hold true? What I would wish for is the following: There is some constant $c$, such that for every finite simple group $G$ there exists a set $\Omega$ with $|\Omega|<(\log G)^c$, on which $G$ acts transitively, such that every subgroup $U$ of $G$, which also acts transitively, is equal to $G$.

Unfortunately this statement is wrong, as can be seen e.g. for $G=\mathrm{Sl}_2(\mathbb{F}_p)$. If $q$ is the smallest prime divisor of $p^2-1$, then there cannot be a set $\Omega$ of size $<q$ with a non-trivial $G$-action, since the stabilizer of a point would be a subgroup of index equal to the size of the orbit of this point.

But what about $\mathrm{Sl}_n(\mathbb{F}_p)$ with $p$ fixed and $n\rightarrow\infty$? What about other sequences of groups?

Thank you in advance!

maximalfactorization of every almost simple group. A quick glance suggests that (for $n$ large enough), there are no factorizations of any classical group with $P_2$, the "second" parabolic group.... $\endgroup$