reference on classfication of multiply transitive permutation groups It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. Is there a canonical reference?
 A: Here is a list of the finite $3$-transitive groups, derived by looking through the list of $2$-transitive groups in Section 7.7 of Dixon and Mortimer and identifying those that are $3$-transitive.
Let's first recall the structure of $G := P{\Gamma}L(2,q)$ with $q=p^e$, $p$ prime. Let $S={\rm PSL}(2,q)$. For $q$ even, $G = S \rtimes \langle \phi \rangle$ with $\phi$ acting as field automorphism of order $e$, and $G/S \cong C_e$.
For $q$ odd, $G = S\langle \delta,\phi \rangle$, where $\delta$ acts as a diagonal automorphism of order $2$, and $\phi$ as a field automorphism, (Note that this extension is nonsplit when $e$ is even.) We have $G/S \cong C_2 \times C_e$. The subgroup $ S \langle \phi \rangle$ of index $2$ in $G$ is denoted by $P{\Sigma}L(2,q)$.
So now, the finite $3$-transitive groups are as follows.
$A_n$ ($n \ge 5$), $S_n$ ($n \ge 3$), degree $n$. (There are two inequivalent actions, conjugate under an outer automorphism of $S_6$,  when $n=6$.)
$A{\Gamma}L(n,2) = {\mathbb F}_2^n \rtimes {\rm GL}(n,2)$ with $n \ge 2$. degree $2^n$.
${\mathbb F}_2^4 \rtimes A_7$, degree $16$.
Groups $G$ with ${\rm PSL}(2,2^e) \le G \le P{\Gamma}L(2,2^e)$, degree $2^n+1$.
For $q$ odd, groups $G$ with ${\rm PSL}(2,q) \le G \le P{\Gamma}L(2,q)$ and $G \not\le P{\Sigma}L(2,q)$, degree $q+1$.
The Mathieu groups $M_{11},M_{12},M_{22},M_{22}.2 = {\rm Aut}(M_{22}), M_{23},M_{24}$, degrees $11,12,22,22,23,24$.
$M_{11}$, degree $12$.
For completeness, the finite $4$-transitive groups are:
$A_n$ ($n \ge 6$), $S_n$ ($n \ge 4$), $M_{11},M_{12},M_{23},M_{24}$,.
The $5$-transitive groups are:
$A_n$ ($n \ge 7$), $S_n$ ($n \ge 5$), $M_{12},M_{24}$.
And the finite $k$-transitive groups for $k \ge 6$ are:
$A_n$ ($n \ge k+2$), $S_n$ ($n \ge k$).
A: This is a bit absurd as a reference, but if you just want an explicit list that has the dubious merit of having gotten past a referee, the $3$-transitive groups are listed on pp. 86-87 of Abhyankar's paper Galois theory on the line in nonzero characteristic.  That list also explicitly states which groups in the list are $4$-transitive, $5$-transitive, etc.  On the other hand, a drawback of Abhyankar's list is that it's missing some of the groups between $\text{PSL}_2(q)$ and $\text{P}\Gamma\text{L}_2(q)$ when $q$ is an odd fourth power.  Personally I prefer the list in Derek Holt's answer, since I value correctness more than publishedness.
