outer automorphisms of $S_6$ $$
\begin{array}{|l|c|c|}
\hline
\text{cycle structure} & \text{number of permutations} & \text{order} \\
\hline
6 & 120 & 6 \\
5+1 & 144 & 5 \\
4+2 & 90 & 4 \\
4+1+1 & 90 & 4 \\
3+3 & 40 & 3 \\
3 + 2 + 1 & 120 & 6 \\
3 + 1 + 1 + 1 & 40 & 3 \\
2 + 2 + 2 & 15 & 2 \\
2 + 2 + 1 + 1 & 45 & 2 \\
2 + 1 + 1 + 1 + 1 & 15 & 2 \\
1 + 1 + 1 + 1 + 1 + 1 & 1 & 1 \\   \hline
\end{array}
$$
In trying to understand outer automorphisms of $S_6$ at the most concrete level of arithmetic, it appears to me that the $15$ generators whose cycle structure is $2+1+1+1+1$ get mapped to the $15$ permutations whose structure is $2+2+2$ (which also generate the group).  And the $40$ permutations whose structure is $3+1+1+1$ similarly get interchanged with the $40$ whose structure is $3+3$; likewise the $120$ whose structure is $3+2+1$ with the $120$ whose structure is $6$.
But apparently the set of $90$ whose structure is $4+2$ is invariant, as is the set of $90$ whose structure is $4+1+1$.


*

*Is this in accord with the experience of those who've actually thought about this?

*Does some web page somewhere walk through this?  I would think the routine arithmetic would have been figured out.


I think the best source for understanding the outer automorphisms of $S_6$ is the following paper:

Combinatorial Structure of the automorphism group of $\mathbf{S_6}$ by T.Y. Lam and David B. Leep, Expositiones Mathematicae 11 (1993), no. 4, pp. 289-903.

It describes exactly how to think about the exchange map between $2$-cycles and "pseudo-transpositions" $(ab)(cd)(ef)$, and it gives nice ways to interpret them combinatorially. I highly recomment it.
