[GAP Forum] list of small groups

[Alexander Konovalov]

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 
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