[GAP Forum] Question

[Stefan Kohl]

Dear Forum,

Katayoon Mehrabadi wrote:

> I want to get some information about Sporadic groups,
> for example conjugacy classes, subgroups etc.

The standard source for information about these groups is
The ATLAS of Finite Groups.

See http://web.mat.bham.ac.uk/R.A.Wilson/atlas.html.

> How can I find their names and symbols which are
> welldefined for GAP. (I know that GAP has
> CharacterTable("co1") but I want to work with it as a
> Group.)

The first question to ask at this point is whether you really need
to work with an explicit representation of this group.

Note that the first Conway group is not quite small, and that many
out-of-the-box methods in GAP will not work for this group, or will
at least require too much time or memory.

If you really need explicit representations of Co1, you can find them
in the ATLAS of Finite Group Representations.
See http://web.mat.bham.ac.uk/atlas/v2.0/.

This database can be accessed via the GAP package AtlasRep, see

http://www.gap-system.org/Packages/atlasrep.html.

If you load this package, you can get a list of stored representations
of Co1 as follows:

   gap> DisplayAtlasInfo("Co1");
   Representations for G = Co1:    (all refer to std. generators 1)
   ----------------------------
    1: G <= Sym(98280)
    2: G <= GL(24,2)
    3: G <= GL(274,2)
    4: G <= GL(276,3)
    5: G <= GL(298,3)
    6: G <= GL(276,5)
    7: G <= GL(299,5)
    8: G <= GL(276,7)
    9: G <= GL(299,7)
   10: G <= GL(276,11)
   11: G <= GL(299,11)
   12: G <= GL(276,13)
   13: G <= GL(299,13)
   14: G <= GL(276,23)
   15: G <= GL(299,23)

   Straight line programs for G = Co1:    (all refer to std. generators 1)
   -----------------------------------
   available maxes of G:  [ 1 .. 6 ]
   repres. of cyclic subgroups of G available
   find standard generators for:  1
   check standard generators for:  1

You can get generators corresponding to the second-mentioned
representation via AtlasGenerators("Co1",2);
This is probably the most `handy' representation of this group.
You can find explicit generators at the end of this mail.

Faithful permutation representations of several smaller sporadic
simple groups can also be obtained by asking for underlying groups
of tables of marks, e.g.

gap> UnderlyingGroup(TableOfMarks("HS"));
<permutation group of size 44352000 with 2 generators>

This currently holds for 2F4(2)', M11, M12, M22, M23, M24,
J1, J2, J3, Co3, HS, McL and He.

The Mathieu groups are also available via MathieuGroup( <degree> ).

Hope this helps,

     Stefan Kohl

- I would like to acknowledge hints by
   Thomas Breuer, Steve Linton and Frank Lübeck -

------------------------------------------------------------------

Standard generators for Co1:

a := [
[0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,1,1,1,1,0,1,1],
[0,1,0,0,1,0,0,1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0,1],
[0,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,1,0,1,1,1,1],
[1,1,0,0,0,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0],
[0,1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,1,0,1,1,0],
[1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,1],
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,1,1,1,1,0,1,0],
[0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,0,1,0,0,0,0,0,0,1],
[0,1,0,1,0,0,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,1,0,1],
[1,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,1,1,1,0,0,1],
[1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,1,0],
[1,0,1,0,0,1,1,1,1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,1],
[1,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0],
[1,0,0,1,0,1,1,1,1,0,1,1,1,0,1,1,1,1,0,1,0,1,1,1],
[1,0,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,0,0,0],
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,1,1,0,1,0,0,1,1],
[0,0,0,1,1,1,1,0,0,0,1,0,0,0,0,1,1,1,0,1,1,0,0,1],
[1,0,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1,1,1,0,1,1,0,0],
[0,1,0,0,1,0,1,0,1,1,1,0,0,1,1,1,0,1,0,1,0,1,1,1],
[0,0,0,0,1,0,1,1,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1],
[0,0,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1],
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0,1,1,0],
[0,1,0,1,0,1,1,1,0,0,1,1,1,0,1,1,0,1,0,1,1,1,0,1],
[0,1,0,0,0,1,0,0,0,0,0,0,1,1,0,1,1,1,1,1,1,1,0,1]
]*Z(2);

b := [
[1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0,1,0,0,0,0,1,1,0],
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,1,1,0,0,0,0,0,1,1,1],
[0,0,0,1,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,1],
[1,1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,0,0,0,1,1,0],
[1,1,0,0,0,0,1,0,0,0,1,0,0,1,0,1,1,1,0,0,1,1,1,0],
[1,0,0,1,0,0,1,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,1],
[0,0,1,0,1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,1,1,1,1,1],
[0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,1,1,0,1,0,1,1,1],
[0,0,0,0,1,1,1,0,0,1,1,0,1,1,0,1,1,1,1,1,0,1,1,0],
[0,0,1,0,1,1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0],
[0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,0,0,1],
[1,1,0,1,1,0,0,0,0,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0],
[0,0,1,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,1,0],
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0,1,0,1,0,0],
[0,1,0,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,0,1,1,1,0,1],
[1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,1,0,1,0,0,0,0,1,0],
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0,1,1],
[1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,1,0,1,0],
[1,1,0,1,0,0,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,1,1],
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,1,1,1,1,1,1],
[0,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],
[0,1,1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,0,0,1,0,1,0,0],
[1,0,0,1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0,0],
[1,0,1,0,0,1,0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,1]
]*Z(2);