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$ \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.
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    I just wrote about it here : http://settheory.net/Out(S6)2018-03-16

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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.

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    Any idea where I can get a copy of the article? The journal's website only seems to provide the articles from 2001 on...2016-08-22