Dear Vahid,
There is a function SmallGroupsInformation that will be helpful to answer this
question.
You will see that the ordering may vary dependently on the order of the group,
for example:
gap> SmallGroupsInformation(256);
There are 56092 groups of order 256.
They are sorted by their ranks.
1 is cyclic.
2 - 541 have rank 2.
542 - 6731 have rank 3.
6732 - 26972 have rank 4.
26973 - 55625 have rank 5.
55626 - 56081 have rank 6.
56082 - 56091 have rank 7.
56092 is elementary abelian.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.
gap> SmallGroupsInformation(105);
There are 2 groups of order 105.
1 is of type 7:3x5.
2 is of type c105.
The groups whose order factorises in at most 3 primes
have been classified by O. Hoelder. This classification is
used in the SmallGroups library.
This size belongs to layer 1 of the SmallGroups library.
IdSmallGroup is available for this size.
gap> SmallGroupsInformation(2*5*7*9);
There are 32 groups of order 630.
They are sorted by their Frattini factors.
1 has Frattini factor [ 210, 1 ].
2 has Frattini factor [ 210, 2 ].
3 has Frattini factor [ 210, 3 ].
4 has Frattini factor [ 210, 4 ].
5 has Frattini factor [ 210, 5 ].
6 has Frattini factor [ 210, 6 ].
7 has Frattini factor [ 210, 7 ].
8 has Frattini factor [ 210, 8 ].
9 has Frattini factor [ 210, 9 ].
10 has Frattini factor [ 210, 10 ].
11 has Frattini factor [ 210, 11 ].
12 has Frattini factor [ 210, 12 ].
13 - 32 have trivial Frattini subgroup.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
LGLength, FrattinifactorSize and FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.
gap>
Best regards,
Alexander
On 13 Aug 2010, at 23:50, Vahid Dabbaghian wrote:
>
> Dear GAP Forum,
>
> Does anybody know what type of ordering is used in the list of small groups
> given by the function AllSmallGroups?
>
> Regards
> Vahid
> ____________________________________________
> Program Director
> Modelling of Complex Social Systems (MoCSSy)
> The IRMACS Centre (ASB 10905)
> Simon Fraser University, BC Canada
> Tel: 778-782-7854
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>
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