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#############################################################################
##
#W smlinfo.gi GAP group library Hans Ulrich Besche
## Bettina Eick, Eamonn O'Brien
##
## This file contains the ...
##
Revision.smlinfo_gi :=
"@(#)$Id: smlinfo.gi,v 4.5 2000/01/06 10:21:53 gap Exp $";
#############################################################################
##
#F SMALL_GROUPS_INFORMATION
##
## ...
SMALL_GROUPS_INFORMATION := [ ];
#############################################################################
##
#F SmallGroupsInformation( size )
##
## ...
InstallGlobalFunction( SmallGroupsInformation, function( size )
local smav, idav, num, lib, t;
smav := SMALL_AVAILABLE( size );
idav := ID_AVAILABLE( size );
if size = 1024 then
Print( "The groups of size 1024 are not available. \n");
return;
fi;
if smav = fail then
Print( "The groups of size ", size, " are not available. \n");
return;
fi;
lib := 1;
if IsBound( smav.lib ) then
lib := smav.lib;
fi;
if IsBound( smav.number ) then
num := smav.number;
else
num := NUMBER_SMALL_GROUPS_FUNCS[ smav.func ]( size, smav ).number;
fi;
if num = 1 then
Print("\n There is 1 group of order ",size,".\n");
else
Print("\n There are ",num," groups of order ",size,".\n" );
fi;
SMALL_GROUPS_INFORMATION[ smav.func ]( size, smav, num );
Print("\n This size belongs to layer ",lib,
" of the SmallGroups library. \n");
if idav <> fail then
Print(" IdGroup is available for this size. \n \n");
else
Print(" IdGroup is not available for this size. \n \n");
fi;
end );
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 1 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 1 ] := function( size, smav, num )
local all, i;
all := AllSmallGroups( size );
for i in [ 1 .. Length( all ) ] do
if HasNameIsomorphismClass( all[ i ] ) then
Print(" ",i," is of type ",NameIsomorphismClass(all[i]),".\n");
else
if HasNameIsomorphismClass( all[ i - 1 ] ) then
Print( " ", i, " - ", Length(all)-1, " are of types " );
if smav.func = 6 then
Print( smav.q,":",smav.p,"+",smav.q,":",smav.p,".\n" );
else
Print( smav.q,":",smav.p,"+",smav.r,":",smav.p,".\n" );
fi;
fi;
fi;
od;
Print("\n");
Print(" The groups whose order factorises in at most 3 primes \n");
Print(" have been classified by O. Hoelder. This classification is \n");
Print(" used in the SmallGroups library. \n");
end;
SMALL_GROUPS_INFORMATION[ 2 ] := SMALL_GROUPS_INFORMATION[ 1 ];
SMALL_GROUPS_INFORMATION[ 3 ] := SMALL_GROUPS_INFORMATION[ 1 ];
SMALL_GROUPS_INFORMATION[ 4 ] := SMALL_GROUPS_INFORMATION[ 1 ];
SMALL_GROUPS_INFORMATION[ 5 ] := SMALL_GROUPS_INFORMATION[ 1 ];
SMALL_GROUPS_INFORMATION[ 6 ] := SMALL_GROUPS_INFORMATION[ 1 ];
SMALL_GROUPS_INFORMATION[ 7 ] := SMALL_GROUPS_INFORMATION[ 1 ];
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 8 .. 10 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 8 ] := function( size, smav, num )
local ffid, prop, i, l;
ffid := IdGroup( OneSmallGroup( size, FrattinifactorSize, size ) );
prop := PROPERTIES_SMALL_GROUPS[ size ].frattFacs;
if not IsPrimePowerInt( size ) then
Print(" There are sorted by their Frattini factors. \n");
i := 1;
repeat
if prop.pos[ i ][ 1 ] = -prop.pos[ i ][ 2 ] then
Print( " ", prop.pos[ i ][ 1 ],
" has Frattini factor ", prop.frattFacs[ i ], ".\n" );
else
Print( " ", prop.pos[ i ][ 1 ], " - ",
-prop.pos[ i ][ 2 ], " have Frattini factor ",
prop.frattFacs[ i ], ".\n" );
fi;
i := i + 1;
until prop.frattFacs[ i ] = ffid;
Print(" ", ffid[2], " - ", num,
" have trivial Frattini subgroup.\n");
else
Print(" There are sorted by their ranks. \n");
Print(" ", 1, " is cyclic. \n");
i := 2;
repeat
l := Length( Factors( prop.frattFacs[ i ][1] ) );
if prop.pos[ i ][ 1 ] = -prop.pos[ i ][ 2 ] then
Print( " ", prop.pos[ i ][ 1 ], " has rank ", l, ".\n" );
else
Print( " ", prop.pos[ i ][ 1 ], " - ",
-prop.pos[ i ][ 2 ], " have rank ", l, ".\n" );
fi;
i := i + 1;
until prop.frattFacs[ i ] = ffid;
Print(" ", ffid[2], " is elementary abelian. \n");
fi;
Print( "\n For the selection functions the values of the ",
"following attributes \n are precomputed and stored:\n ");
if IsPrimePowerInt( size ) then
Print( " IsAbelian, PClassPGroup, RankPGroup,",
" FrattinifactorSize and \n FrattinifactorId. \n");
else
Print( " IsAbelian, IsNilpotentGroup,",
" IsSupersolvableGroup, IsSolvableGroup, \n LGLength,",
" FrattinifactorSize and FrattinifactorId. \n");
fi;
end;
SMALL_GROUPS_INFORMATION[ 9 ] := SMALL_GROUPS_INFORMATION[ 8 ];
SMALL_GROUPS_INFORMATION[ 10 ] := SMALL_GROUPS_INFORMATION[ 8 ];
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 11, 17 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 11 ] := function( size, smav, num )
local i, q;
q := 2;
if IsBound( smav.q ) then q := smav.q; fi;
Print(" There are sorted by normal Sylow subgroups. \n");
Print( " 1 - ", smav.pos[ 2 ], " are the nilpotent groups.\n" );
for i in [ 2 .. Length( smav.types ) ] do
Print( " ", smav.pos[i] + 1, " - ", smav.pos[i+1] );
if smav.types[ i ] = "p-autos" then
Print( " have a normal Sylow ", q,"-subgroup. \n");
elif smav.types[ i ] = "none-p-nil" then
Print( " have no normal Sylow subgroup. \n");
elif IsInt( smav.types[ i ] ) then
Print( " have a normal Sylow ", smav.p, "-subgroup \n");
Print( " with centralizer of index ");
Print( q^smav.types[i],".\n");
fi;
od;
end;
SMALL_GROUPS_INFORMATION[ 17 ] := SMALL_GROUPS_INFORMATION[ 11 ];
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 12 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 12 ] := function( size, smav, num )
if size = 1152 then
Print(" There are sorted using Sylow subgroups. \n");
Print(" 1 - 2328 are nilpotent with Sylow 3-subgroup c9.\n" );
Print(" 2329 - 4656 are nilpotent with Sylow 3-subgroup 3^2.\n");
Print(" 4657 - 153312 are non-nilpotent with normal ");
Print("Sylow 3-subgroup.\n");
Print(" 153313 - 157877 have no normal Sylow 3-subgroup.\n");
return;
fi;
Print(" There are sorted using Hall subgroups. \n");
Print( " 1 - 2328 are the nilpotent groups.\n" );
Print( " 2329 - 236344 have a normal Hall (3,5)-subgroup.\n");
Print( " 236345 - 240416 are solvable without normal Hall",
" (3,5)-subgroup.\n");
Print( " 240417 - 241004 are not solvable.\n" );
end;
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 14 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 14 ] := function( size, smav, num )
Print( " 1 - 10494213 are the nilpotent groups.\n" );
Print( " 10494214 - 408526597 have a normal Sylow 3-subgroup.\n" );
Print( " 408526598 - 408544625 have a normal Sylow 2-subgroup.\n" );
Print( " 408544626 - 408641062 have no normal Sylow subgroup.\n" );
end;
#############################################################################
##
#F SMALL_GROUPS_INFORMATION[ 18 ]( size, smav, num )
##
SMALL_GROUPS_INFORMATION[ 18 ] := function( size, smav, num )
Print( " 1 is cyclic. \n");
Print( " 2 - 10 have rank 2 and p-class 3.\n" );
Print( " 11 - 386 have rank 2 and p-class 4.\n" );
Print( " 387 - 1698 have rank 2 and p-class 5.\n" );
Print( " 1699 - 2008 have rank 2 and p-class 6.\n" );
Print( " 2009 - 2039 have rank 2 and p-class 7.\n" );
Print( " 2040 - 2044 have rank 2 and p-class 8.\n" );
Print( " 2045 has rank 3 and p-class 2.\n" );
Print( " 2046 - 29398 have rank 3 and p-class 3.\n" );
Print( " 29399 - 56685 have rank 3 and p-class 4.\n" );
Print( " 56686 - 60615 have rank 3 and p-class 5.\n" );
Print( " 60616 - 60894 have rank 3 and p-class 6.\n" );
Print( " 60895 - 60903 have rank 3 and p-class 7.\n" );
Print( " 60904 - 67612 have rank 4 and ", "p-class 2.\n" );
Print( " 67613 - 387088 have rank 4 and ", "p-class 3.\n" );
Print( " 387089 - 419734 have rank 4 and ", "p-class 4.\n" );
Print( " 419735 - 420500 have rank 4 and ", "p-class 5.\n" );
Print( " 420501 - 420514 have rank 4 and ", "p-class 6.\n" );
Print( " 420515 - 6249623 have rank 5 and ", "p-class 2.\n" );
Print( " 6249624 - 7529606 have rank 5 and ", "p-class 3.\n" );
Print( " 7529607 - 7532374 have rank 5 and ", "p-class 4.\n" );
Print( " 7532375 - 7532392 have rank 5 and ", "p-class 5.\n" );
Print( " 7532393 - 10481221 have rank 6 and ", "p-class 2.\n" );
Print( " 10481222 - 10493038 have rank 6 and ", "p-class 3.\n" );
Print( " 10493039 - 10493061 have rank 6 and ", "p-class 4.\n" );
Print( " 10493062 - 10494173 have rank 7 ", "and p-class 2.\n" );
Print( " 10494174 - 10494200 have rank 7 ", "and p-class 3.\n" );
Print( " 10494201 - 10494212 have rank 8 ", "and p-class 2.\n" );
Print( " 10494213 is elementary abelian.\n");
end;
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