In a comment at the recent question What is the standard 2-generating set of the symmetric group good for?, it was remarked that the symmetric groups $S_n$ for $n\gt 2$, $n\neq 5,6,8$, can be generated by a transposition and an element of order 3 (G. A. Miller, Bull. Amer. Math. Soc. 7 (1901), 424-426 doi:10.1090/S0002-9904-1901-00826-9). The remaining three nonabelian cases can of course be generated by a pair of elements, but these are cycles of length $5,6,8$ respectively. What is the best that can be done in these cases, and is there a conceptual reason why these are exceptional? (eg the presence of the nontrivial outer automorphism of $S_6$? Or some action on an exceptional combinatorial object?)