[GAP Forum] Permutation Representation of 3.McL.2

[Stephen Linton]

Dear Joe, dear GAP Forum,

There is a standard trick for this situation, which comes in two  
parts, depending on whether the 66k point perm rep is taken to an  
equivalent or an inequivalent representation by the outer automorphism  
of 3.Mcl

In the inequivalent case, it's very easy. Call the original  
representation rho and the image under the outer automorphism rho'.
You take twice as many points, with 3McL acting as rho on one half and  
rho' on the other. Adjoin an involution exchanging the two halves and  
you're done.

In the equivalent case you have to do a little more work. You find  
words for a transversal of the points in your standard generators  
w_1,....w_66k
Now you make the same words in the images of the generators under the  
automorphism, call the resulting set of group elements x_1..x_66k.
The permutation which maps 1 to 1.x_1, 2 to 1.x_2, 3 to 1.x_3 and so  
on extends the representation to 3 McL.2.

	Steve

On 5 May 2009, at 17:42, Joe Bohanon wrote:

> Does anyone know a good way to get a permrep for 3.McL.2 (the second  
> largest subgroup of the Lyons group)?  The ATLAS has a rep for 3.McL  
> with about 66k moved points, so I'm guessing 3.McL.2 is probably not  
> much bigger than that.
>
> What I really need is a list of the classes of non-solvable  
> subgroups of 3.McL.2.  I could theoretically do this by creating a  
> homomorphism to McL.2, then lifting non-solvable subgroups and  
> testing if the "3" is complemented or not in the pre-images.  I can  
> easily create the homomorphism using GroupHomomorphismByImagesNC,  
> but GAP stalls bad when trying to take pre-images.
>
> I've thought about using the ideas from the paper "Constructing  
> Permutation Representations for Large Matrix Groups", but before I  
> embark on trying to encode their algorithm, I figured I'd ask here  
> to see if anyone might just happen to have generators you could send  
> me.
>
> Thanks
> Joe
>
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Steve Linton	School of Computer Science  &
       Centre for Interdisciplinary Research in Computational Algebra
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