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Some time back I posted some questions about the "exotic" outer automorphisms of $S_6$, and part of the answer was a citation of a paper by T. Y. Lam that said, among other things, that the group of all automorphisms of $S_6$ has exactly three subgroups of index $2$, and the other two are not isomorphic either to the group of inner automorphisms or to each other.

One might think that exhausts the list of "exotic" things found among permutation groups acting on finite sets, since it is said that no other finite symmetric group has any outer automorphisms.

Later I find (doubtless group-theorists have known this since toddlerhood and will be shocked that some first learn of it only after getting their driver's license) that there exist six isomorphisms from $S_5$ into $S_6$ whose images are conjugate to each other and act transitively on the set of six objects permuted by members of $S_6$.

So my question is: Which other "exotic" objects are to be found among groups of bijections of finite sets? And how should the word "exotic" be understood (might there actually be a precise definition)?

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Michael Hardy
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    Your second fact corresponds simply to the fact that the normalizer of the Sylow $5$-subgroup in $S_5$ has index $6$, no? – Mariano Suárez-Álvarez May 19 '15 at 21:47
  • @MarianoSuárez-Alvarez : I think so. – Michael Hardy May 19 '15 at 21:50
  • I apologize, this is very basic, and I don't doubt you are aware of it, but it relates to $S_4$. There are 3 ways to cut a 4 element set in two equal halves, hence there exists a (clearly) surjective morphism $S_4\to S_3$. – Olivier Bégassat May 19 '15 at 21:54
  • @OlivierBégassat : Is the thing that makes this homomorphism "exotic" the fact that the things being permuted by this realization of $S_3$ are not the things being permuted by $S_4$? (Instead, they are cosets of a subgroup of $S_4$.) ${}\qquad{}$ – Michael Hardy May 26 '15 at 02:18

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In some sense of exotic $S_6$ is unique. In a completely relaxed interpretation of exotic, any group with a nontrivial outer automorphism will fit the bill. Attempting to find a happy medium, perhaps you'd be interested studying something like the simple group of order $168$. It has a permutation representation on $7$ points. The stabilizer of a point is isomorphic to $S_4$ and so there are $7$ subgroups isomorphic to $S_4$. But wait there's more. There are $7$ other subgroups also isomorphic to $S_4$ which are not point stablilizer (they don't act transitively on $7$ points, but on subsets of size $3$ and $4$.) And indeed there is an outer automorphism which maps the two sets to each other.

Josh B.
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