# [GAP Forum] Obtaining Small Group information

Joe Bohanon jbohanon2 at gmail.com
Fri Dec 12 07:01:38 GMT 2008

```Also in general the groups of order p^3*q were classified by Western
near the beginning of the 20th century (if you have Burnside's book, you
can find the reference there where he does p^2 q).  I think very few
generic classifications were done beyond what Hoelder did (except for
the work on p-groups by O'Brien and Eick) mainly because of the number
of sub-cases you have to look at.

Heiko Dietrich wrote:
> Hello,
>
> the groups of order 5^3 are the following
>
> (1) cyclic group C_{125}
> (2) abelian group C_{25} x C_5
> (3) extraspecial 5-group of order 125 and exponent 5:  C_5 \ltimes (C_5 x C_5)
> (4) extraspecial 5-group of order 125 and exponent 25:  C_5 \ltimes C_{25}
> (5) elementary abelian group C_5 x C_5 x C_5.
>
> (Note that A \ltimes B is a split extensions with A acting on the normal
> subgroup B.)
>
> In general, the groups of order p^3 were classified by Otto Hoelder, "Die
> Gruppen der Ordnungen p^3 , pq^2, pqr, p^4.", Math. Ann., 43: 301 - 412,
> 1893.
>
> As 1625 = 13*5^3, a group G of order 1625 has a normal Sylow subgroup P of
> order 125. Now Schur-Zassenhaus show that P has a complement in G, that is, G
> is isomorphic to the split extension C_{13} \ltimes P. Due to order reasons,
> the cyclic group has to act trivially, that is, G is isomorphic to the direct
> product C_{13}  x P.
>
> Hence, you obtain all groups of order 1625 by adding a direct factor C_{13} to
> every group (1)--(5) of order 125:
>
> gap>NumberSmallGroups(1625);
> 5
> gap>List(AllSmallGroups(125),x->IdSmallGroup(DirectProduct(x,CyclicGroup(13))));
> [ [ 1625, 1 ], [ 1625, 2 ], [ 1625, 3 ], [ 1625, 4 ], [ 1625, 5 ] ]
>
> Hope this helps,
> Heiko
>
>
> On Thursday 11 December 2008 23:19, Paweł Laskoś-Grabowski wrote:
>
>> 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,
>>
>> 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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>
>

```