Fw: [GAP Forum] Listing Octads

[Paul_Hjelmstad at allianzlife.com]

Alexander,

Thanks. This works, except that I get a different value for the first 
octad (orbit of 2-Sylow group), it lists a 16-chord first, and then, 
Set(octad)[1] ends
up being a 16-chord, and so do all the others. Do you know why it is 
listing a 16-chord first, and why my octad is different? I get

 [ 3, 14, 22, 15, 11, 23, 17, 19 ]

Paul Hjelmstad
ITS Security Administration
763-582-6547 (x46547)
Paul_Hjelmstad at allianzlife.com
----- Forwarded by Paul Hjelmstad/allianzlife on 03/27/2007 11:40 AM -----

"PAUL HJELMSTAD" <phjelmstad at msn.com> 
03/01/2007 03:08 PM

To
paul_hjelmstad at allianzlife.com
cc

Subject
FW: Re: [GAP Forum] Listing Octads









>From: Alexander Hulpke <hulpke at math.colostate.edu>
>To: "PAUL HJELMSTAD" <phjelmstad at msn.com>
>CC: forum at gap-system.org
>Subject: Re: [GAP Forum] Listing Octads
>Date: Thu, 1 Mar 2007 11:00:39 -0700
>
>Dear GAP forum,
>
>Paul Hjelmstad wrote:
>
>>Does anyone know how I would list:
>Yes.
>
>>2) Representatives of Conjugacy Classes (26) in M24
>gap> g:=MathieuGroup(24);
>gap> List(ConjugacyClasses(g),Representative);
>(Careful: Order is not necessarily ATLAS order but its easy to get a 
>correspondence)
>
>>1) Octads in M24 (Are they Unique?)
>The 759 octads are one orbit under M24 and are determined uniquely by  5 
>points of this 5-transitive group, thus there are unique.
>
>As the index of an octad stabilizer is 759 and thus odd, we can get  one 
>octad from orbits of a 2-Sylow subgroup:
>
>gap> s:=SylowSubgroup(g,2);
><permutation group of size 1024 with 10 generators>
>gap> Orbits(s,[1..24]);
>[ [ 1, 5, 24, 8, 10, 11, 14, 20 ],
>   [ 2, 13, 22, 19, 7, 3, 9, 23, 18, 4, 15, 12, 6, 16, 17, 21 ] ]
>gap> o:=Set(last[1]);
>
>The set of octads then is obtained as orbit:
>
>gap> octads:=Orbit(g,o,OnSets);;
>gap> Length(octads);
>759
>
>from which we see that in this labelling of M24 the lexicographically 
>smallest octad is
>
>gap> Set(octads)[1];
>[ 1, 2, 3, 4, 5, 8, 11, 13 ]
>
>
>Best,
>
>    Alexander Hulpke
>
>
>-- Colorado State University, Department of Mathematics,
>Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
>email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
>http://www.math.colostate.edu/~hulpke
>
>




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