73.60 Cat1Select

Cat1Select( size, [gpnum, num] )

All cat-structures on groups of order up to 47 are stored in a list Cat1List and may be obtained from the list using this function. Global variables Cat1ListMaxSize := 47 and NumbersOfIsomorphismClasses are also stored. The example illustrated is the first case in which t ne h and the associated conjugation crossed module is given by the normal subgroup c3 of s3.

    gap> Cat1ListMaxSize;
    47
    gap> NumbersOfIsomorphismClasses[18];
    5
    gap> Cat1Select( 18 );
    Usage:  Cat1Select( size, gpnum, num )
    [ "c6c3", "c18", "d18", "s3c3", "c3^2

Xc2" ]

    gap> Cat1Select( 18, 5 );
    There are 4 cat1-structures for the group c3^2

Xc2.
    [ [range generators], [tail.genimages], [head.genimages] ]  :- 
    [ [ (1,2,3), (4,5,6), (2,3)(5,6) ],  tail = head = identity mapping ]
    [ [ (2,3)(5,6) ], "c3^2", "c2", [ (), (), (2,3)(5,6) ],
        [ (), (), (2,3)(5,6) ] ]
    [ [ (4,5,6), (2,3)(5,6) ], "c3", "s3", [ (), (4,5,6), (2,3)(5,6) ], 
        [ (), (4,5,6), (2,3)(5,6) ] ]
    [ [ (4,5,6), (2,3)(5,6) ], "c3", "s3", [ (4,5,6),(4,5,6),(2,3)(5,6) ],
        [ (), (4,5,6), (2,3)(5,6) ] ]
    Usage:  Cat1Select( size, gpnum, num )
    Group has generators [ (1,2,3), (4,5,6), (2,3)(5,6) ]
 
    gap> SC := Cat1Select( 18, 5, 4 );
    cat1-group [c3^2

Xc2 ==> s3] 
    gap> Cat1Print( SC );

    cat1-group [c3^2

Xc2 ==> s3] :- 
    : source group has generators:
      [ (1,2,3), (4,5,6), (2,3)(5,6) ]
    :  range group has generators:
      [ (4,5,6), (2,3)(5,6) ]
    : tail homomorphism maps source generators to:
      [ ( 4, 5, 6), ( 4, 5, 6), ( 2, 3)( 5, 6) ]
    : head homomorphism maps source generators to:
      [ (), ( 4, 5, 6), ( 2, 3)( 5, 6) ]
    : range embedding maps range generators to:
      [ ( 4, 5, 6), ( 2, 3)( 5, 6) ]
    : kernel has generators:
      [ ( 1, 2, 3)( 4, 6, 5) ]
    : boundary homomorphism maps generators of kernel to:
      [ ( 4, 6, 5) ]
    : kernel embedding maps generators of kernel to:
      [ ( 1, 2, 3)( 4, 6, 5) ]

    gap> XSC := XModCat1( SC );
    Crossed module [c3->s3]   

For each group G the first cat1-structure is the identity cat1-structure (id;id,id : G -> G) with trivial kernel. The corresponding crossed module has as boundary the inclusion map of the trivial subgroup.

    gap> AC := Cat1Select( 12, 5, 1 );
    cat1-group [a4 ==> a4]   

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