‣ SmallClassNrGroup( k, i ) | ( function ) |
‣ SmallClassNrGroup( k, i: AsPermGroup ) | ( function ) |
Returns: the i-th finite group of class number k in the library.
By default, if the group is soluble, it is given as a PcGroup whose Pcgs is a SpecialPcgs. If the group is not soluble, or if the option AsPermGroup is added, it will be given as a permutation group of minimal permutation degree and with a minimal generating set.
gap> SmallClassNrGroup( 4, 4 ); <pc group of size 12 with 3 generators> gap> G := SmallClassNrGroup( 4, 4 : AsPermGroup ); Group([ (1,2,3), (1,4,2) ]) gap> NrConjugacyClasses( G ); 4 gap> IsAlternatingGroup( G ); true
‣ IdClassNr( G ) | ( attribute ) |
Returns: the SmallClassNr ID of G, i.e. a pair [k, i] such that G is isomorphic to SmallClassNrGroup(k, i).
gap> IdClassNr( AlternatingGroup( 4 ) ); [ 4, 4 ]
For each of the functions in this section, the arguments arg must come in pairs consisting of a function and a value (or list of accepted values). At least one of the functions must be NrConjugacyClasses. Missing functions will be interpreted as NrConjugacyClasses, missing values as true.
The option AsPermGroup can be added to the functions in this section to ensure that all groups are returned as PermGroups (instead of PcGroups if they are soluble).
‣ AllSmallClassNrGroups( arg... ) | ( function ) |
‣ AllSmallClassNrGroups( arg...: AsPermGroup ) | ( function ) |
Returns: all finite groups with certain properties as specified by arg.
gap> AllSmallClassNrGroups( IsSolvable, true, NrConjugacyClasses, 6 ); [ <pc group of size 6 with 2 generators>, <pc group of size 12 with 3 generators>, <pc group of size 12 with 3 generators>, <pc group of size 18 with 3 generators>, <pc group of size 18 with 3 generators>, <pc group of size 36 with 4 generators>, <pc group of size 72 with 5 generators> ] gap> AllSmallClassNrGroups( [ 3 .. 5 ], IsNilpotent ); [ <pc group of size 3 with 1 generator>, <pc group of size 4 with 2 generators>, <pc group of size 4 with 2 generators>, <pc group of size 5 with 1 generator>, <pc group of size 8 with 3 generators>, <pc group of size 8 with 3 generators> ] gap> AllSmallClassNrGroups( [ 3 .. 5 ], IsNilpotent : AsPermGroup ); [ Group([ (1,2,3) ]), Group([ (1,2,3,4) ]), Group([ (1,2), (3,4) ]), Group([ (1,2,3,4,5) ]), Group([ (1,2), (1,3)(2,4) ]), Group([ (1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6) ]) ]
‣ OneSmallClassNrGroup( arg... ) | ( function ) |
‣ OneSmallClassNrGroup( arg...: AsPermGroup ) | ( function ) |
Returns: one finite group with certain properties as specified by arg.
gap> OneSmallClassNrGroup( 6, IsSolvable, false ); Group([ (1,2,3)(4,5,6), (1,4)(2,7) ]) gap> OneSmallClassNrGroup( 10, IsSolvable, true, IsNilpotent, false ); <pc group of size 28 with 3 generators> gap> OneSmallClassNrGroup( 10, IsSolvable, true, IsNilpotent, false : AsPermGroup ); Group([ (1,2,3,4,5,6,7), (2,7)(3,6)(4,5)(8,9,10,11) ])
‣ NrSmallClassNrGroups( arg... ) | ( function ) |
Returns: the number of finite groups with certain properties as specified by arg.
gap> NrSmallClassNrGroups( 14 ); 93 gap> NrSmallClassNrGroups( IsSolvable, true, NrConjugacyClasses, 6 ); 7 gap> NrSmallClassNrGroups( [ 3 .. 5 ], IsNilpotentGroup ); 6
‣ IteratorSmallClassNrGroups( arg... ) | ( function ) |
Returns: an iterator that iterates over the finite groups with properties as specified by arg.
gap> iter := IteratorSmallClassNrGroups( 12, IsSimpleGroup ); <iterator> gap> for G in iter do Print( Size( G ), "\n" ); od; 3420 5616 443520
‣ SmallClassNrGroupsAvailable( k ) | ( function ) |
Returns: true if the finite groups of class number k are available in the library, and false otherwise.
gap> SmallClassNrGroupsAvailable( 14 ); true gap> SmallClassNrGroupsAvailable( 15 ); false
generated by GAPDoc2HTML