In a [comment](https://mathoverflow.net/questions/405256/what-is-the-standard-2-generating-set-of-the-symmetric-group-good-for#comment1038737_405273) at the recent question https://mathoverflow.net/questions/405256/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 an element of order 2 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](https://doi.org/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?)