[GAP Forum] Obtaining Small Group information

[Paweł Laskoś-Grabowski]

Hello,

Much of this (highly useful otherwise, thanks a lot) information is 
actually much more general than I need at the moment. I need to know the 
structure of all (up to isomorphism, of course) groups of orders 125 and 
1625. I was glad to discover that there are only five of each, but now 
it seems that the ones obtained by semidirect products may actually 
represent many non-isomorphic groups. Is there a way to obtain such 
level of details using GAP, or should I refer to textbooks and/or prove 
few facts myself to get the information I need?

Regards,
Pawel Laskos-Grabowski

Joe Bohanon schrieb:
> 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 
> <mailto: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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