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

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

Dear forum, Dear Robert,

for what it's worth, here is what I described, in GAP,
takes an instant to run:

LoadPackage("grape");
g0:=AtlasGroup("3.M22.2");
g693:=Image(IsomorphismPermGroup(g));
h:=Stabilizer(g693,1);
oo:=Orbits(h,[1..693]);
e:=[1,First(oo,x->Length(x)=30)[1]];
gamma:=NullGraph(g693);
AddEdgeOrbit(gamma,e);
c7:=Orbit(g693,CompleteSubgraphs(gamma,7)[1],OnSets);
iif:=NullGraph(Action(g693,c7,OnSets));
AddEdgeOrbit(iif,
  List(Filtered(c7,x->IsSubset(x,e)),x->Position(c7,x)));

# Now we call
GlobalParameters(iif);
# and see that it is
# [ [ 0, 0, 7 ], [ 1, 0, 6 ], [ 1, 2, 4 ], [ 1, 2, 4 ],
# [ 2, 1, 4 ], [ 4, 2, 1 ], [ 4, 2, 1 ], [ 6, 0, 1 ], [ 7, 0, 0 ] ] 
# - the parameters of Ivanov-Ivanov-Faradjev graph

Cheers,
Dima

On Tue, Nov 15, 2016 at 08:51:17PM +0000, dmitrii.pasechnik at cs.ox.ac.uk wrote:
> 
> 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/
> > 
> > This electronic communication is governed by the terms and conditions at http://www.mun.ca/cc/contact/policies/elect_communications_disclaimer_2011.php
> > _______________________________________________
> > Forum mailing list
> > Forum at mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum