Re: Outer automorphisms of S6 [group of permutations of 6 things]

In article <ecile4$1sdi$1@xxxxxxxxxxxxxxxxxx>,
magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin) writes:
In article <3042278.1156363979927.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Richard L. Peterson <rl_pete@xxxxxxxxx> wrote:

[...]

So we were merely talking at cross
purposes with different meanings in mind.


-----Yes, good that's straightened out.

Now, since S_6 is centerless, of course, Aut(S_6) itself satisfies
your conditions, with Inn(S_6) corresponding to the normal subgroup
isomorphic to S_6. So I guess that what you are hoping for is some
kind of action of Aut(S_6) or a representation of Aut(S_6) that lets
you understand the entire group better.

I honestly confess I cannot help you with that; it has been only
slowly, and at some point almost kicking and screasming, that I've
become a fan of actions and representations as good ways to think
about a group. Very short-sighted of me, to be sure. But as a
consequence, I am not familiar enough with it to give you, say, the
irreducible representations of Aut(S_6). But, presumably, by studying
->those<- you might get the sense of the outer automorphisms that you
want. There is, if I remember correctly, some of that in the article
by Lam.

Here is some information on representations, but I am not sure how useful
it will prove to be!

The smallest degree faithful permutation representation of Aut(S_6) has
degree 10. In fact, Aut(S_6) is isomorphic to the group PGammaL(2,9),
which acts naturally on 10 points. Here are some images under such a
representation:

Element of S_6 image in representation of degree 10
(1,2) --> (1,4)(2,5)(7,9)
(1,2,3,4,5,6) --> (1,2,10,9,8,5)(3,4,6)

and an element phi of Aut(S_6) which interchanges (1,2) and (1,4)(2,5)(3,6)
is represented by the degree 10 permutation (1,5)(2,9)(3,6)(4,10)(7,8).

The irreducible complex representations of Aut(S_6) have degrees:
1,1,1,1,9,9,9,9,10,10,16,16,20.
One of the degree 9 representations arises as a deleted permutation module
from the degree 10 permutation representation.

There is also a representation of degree 6 over the field of order 3.

Derek Holt.

.