[GAP Forum] Obtaining Small Group information

[Joe Bohanon]

I would point out that StructureDescription might not always return a group
the way you'd like it.  The manual explains a little more about how it picks
a particular form for the structure.

That function also does not do anything with central products.  Hence if I
type:
StructureDescription(SmallGroup(32,50)) I get:
"(C2 x Q8) : C2" when it's also a central product of Q8 with D8.  It returns
some pretty awkward answers for other larger central products.

It also will usually not let you know how the split or non-split extensions
work, so you might get two non-isomorphic groups that return the same
"StructureDescription".

Also be forewarned that many times GAP will just compute the whole subgroup
lattice to find a structure, so any group that would take a long time with
LatticeByCyclicExtension or ConjugacyClassesSubgroups is likely to take a
long time for StructureDescription.  This would include, for instance,
2-groups of rank more than 5, groups with large permutation representations
or large matrix representations and also finitely-presented groups.  It does
have a separate routine for any simple group that spits out the answer due
to the classification in almost no time, however, while it could easily tell
me a group is isomorphic to, say U4(3), it would take much longer (and
probably use up all of your RAM) to say a group is isomorphic to U4(3):D8.

On Thu, Dec 11, 2008 at 6:37 AM, Heiko Dietrich <h.dietrich at tu-bs.de> wrote:

> Dear Paweł,
>
> you can use the command "StructureDescription":
>
> gap> for i in AllSmallGroups(1625) do Display(StructureDescription(i)); od;
> C1625
> C325 x C5
> C13 x ((C5 x C5) : C5)
> C13 x (C25 : C5)
> C65 x C5 x C5
>
> The output is explained in the manual:
>
> http://www.gap-system.org/Manuals/doc/htm/ref/CHAP037.htm#SECT006
>
> Best,
> Heiko
>
>
>
> On Tuesday 09 December 2008 20:56, Paweł Laskoś-Grabowski wrote:
> > Hello,
> >
> > I have noticed that GAP Small Groups library provides useful information
> > on the structure of groups belonging to the layer 1 of the library, but
> > does not do so for (some) bit more complicated groups. I am rather
> > dissatisfied by the output
> >
> > gap> SmallGroupsInformation(1625);
> >
> >    There are 5 groups of order 1625.
> >    They are sorted by normal Sylow subgroups.
> >       1 - 5 are the nilpotent groups.
> >
> > How can I obtain such a pleasant info like the following?
> >
> > gap> SmallGroupsInformation(125);
> >
> >    There are 5 groups of order 125.
> >      1 is of type c125.
> >      2 is of type 5x25.
> >      3 is of type 5^2:5.
> >      4 is of type 25:5.
> >      5 is of type 5^3.
> >
> > And, by the way, what does the colon stand for in the 125,3 and 125,4
> > type descriptions? I failed to find the explanation in the help pages.
> >
> > Regards,
> > Paweł Laskoś-Grabowski
> >
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>
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