It seems to be well-known that the six-transitive finite groups are the symmetric and alternating groups, and the only other four-transitive finite groups are the Mathieu groups (the statement can be found in Cameron's 1999 "Permutation Groups", p 110), but I can't seem to figure out where the result was first stated (Cameron is not much help, though he does give a vague sketch of how one might go about proving this result).
In Pacific Journal of Math 4 (1954), pp 219-226, Marshall Hall, Jr. writes in the paper "On a theorem of Jordan" that:
(Hall's paper is available from Project Euclid ).
The citation is: C. Jordan, Recherches sur les substitutions, J. Math. Pures Appl. (2) 17 (1872), 351-363.
Hall generalizes the result to any group (finite or infinite) in which the stabilizer of four letters is finite of odd order. As I recall, he presents the result extending it from 4-transitive to $k$-transitive (with $k\gt 3$) in his book The Theory of Groups, but I'm away from the office this week and unable to verify this.
Peter Cameron has an article "Finite permutation groups and finite simple groups. He states that Wielandt showed that any 6-transitve group is alternating or symmetric, given the Schreier's conjecture is true. Schreier's conjecture is that the outer automorphism group of a finite simple group is solvable, and the truth of this is a consequence of the classification.
I found Cameron's article by googling on "Wielandt Schreier". Wielandt's result was published as H. Wielandt, "Uber den Transitivitatsgrad von Permutationsgruppen", Math. Z. 74 (1960), 297-298.
On page 218 of
John D. Dixon, Brian Mortimer: Permutation Groups, Springer GTM 163, 1996
it is stated:
Though Dixon / Mortimer also do not say where the result has been first stated.