[GAP Forum] 3.M_{22}:2 on 990 points

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear forum, Dear Robert,

On Tue, Nov 15, 2016 at 07:59:48PM +0000, Bailey, Robert F. wrote:
> The group 3.M_{22}:2 has a rank-9 imprimitive permutation representation of degree 990. (This is the full automorphism group of the distance-transitive Ivanov-Ivanov-Faradjev graph.)  I would like to construct this group in GAP.
> 
> The www ATLAS gives a matrix representation of this group in characteristic 2; however, applying "IsomorphismPermGroup" to this matrix group gives a group of degree 693.
> 
> Does anyone have a suggestion for how to obtain the degree 990 representation?

There is a diagram geometry with this group that has 990 points and 693
lines; more concretely, there is a degree 7 graph of girth 5 on the 990 vertices
that is invariant under your group G, such that every 2-path lies in a
unique Petersen subgraph; there are 693 these subgraphs.
Each vertex is in 7 such subgraphs.
Dually, for the permutation representation of degree 693 there is a
G-invariant graph of degree 30, such each vertex lies in 10 maximal
cliques of size 7; there are 990 such cliques, and the action on them
will give you the desired action.

I would use GRAPE to construct the graph on 693 vertices, and find a
7-clique there. Also the following might help:
the maximum possible intersection of two such 7-cliques
is in a 3-clique, and there are 15 such special 3-cliques on each of the
693 vertices- they correspond to the edges of your graph on 990
vertices.

Hope this helps.
I'd be happy to provide more details, if needed.
Dima



> Thanks,
> Robert Bailey.
> 
> ==============================
> Dr. Robert Bailey
> School of Science and Environment (Mathematics)
> Grenfell Campus
> Memorial University of Newfoundland
> Corner Brook, NL A2H 6P9, Canada
> 
> Office: AS 3022
> Phone: +1 (709) 637-6293
> Web: http://www2.grenfell.mun.ca/rbailey/
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