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#############################################################################
##
#W idgrp1.g GAP group library Hans Ulrich Besche
## Bettina Eick, Eamonn O'Brien
##
## This file contains the identification routines for groups with order
## the product of maximal 3 primes.
##
#############################################################################
##
#F ID_AVAILABLE_FUNCS[ 1 ]
##
ID_AVAILABLE_FUNCS[ 1 ] := SMALL_AVAILABLE_FUNCS[ 1 ];
#############################################################################
##
#F ID_GROUP_FUNCS[ 1 ]( G, inforec )
##
## order p
##
ID_GROUP_FUNCS[ 1 ] := function( G, inforec )
return 1;
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 2 ]( G, inforec )
##
## order p ^ 2
##
ID_GROUP_FUNCS[ 2 ] := function( G, inforec )
if IsCyclic( G ) then
return 1;
else
return 2;
fi;
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 3 ]( G, inforec )
##
## order p * q
##
ID_GROUP_FUNCS[ 3 ] := function( G, inforec )
local typ;
if IsAbelian( G ) then
typ := "pq";
else
typ := "Dpq";
fi;
return Position(
NUMBER_SMALL_GROUPS_FUNCS[ 3 ]( Size( G ), inforec ).types, typ );
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 4 ]( G, inforec )
##
## order p ^ 3
##
ID_GROUP_FUNCS[ 4 ] := function( G, inforec )
if IsAbelian( G ) then
if IsCyclic( G ) then
return 1;
elif IsElementaryAbelian( G ) then
return 5;
fi;
return 2;
else
if Size( G ) = 8 then
if Length( Filtered( AsList( G ), x-> Order( x ) = 2 ) ) = 1 then
return 4;
fi;
return 3;
fi;
if Maximum( List( GeneratorsOfGroup( G ), x -> Order( x ) ) ) =
FactorsInt( Size( G ) )[ 1 ] then
return 3;
fi;
return 4;
fi;
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 5 ]( G, inforec )
##
## order p ^ 2 * q
##
ID_GROUP_FUNCS[ 5 ] := function( G, inforec )
local n, p, q, s1, s2, typ;
# get primes
n := Size(G);
p := FactorsInt(n);
q := p[3];
p := p[1];
# compute the sylow subgroups
s1 := SylowSubgroup( G, q );
s2 := SylowSubgroup( G, p );
if IsAbelian( G ) then
if IsCyclic( s2 ) then
typ := "p2q";
else
typ := "ppq";
fi;
elif IsElementaryAbelian( s2 ) then
if n = 12 and IsNormal( G, s2 ) then
typ := "a4";
else
typ := "Dpqxp";
fi;
elif not IsTrivial( Centralizer( s2, s1 ) ) then
typ := "Gp2q";
else
typ := "Hp2q";
fi;
return Position(
NUMBER_SMALL_GROUPS_FUNCS[ 5 ]( Size( G ), inforec ).types, typ );
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 6 ]( G, inforec )
##
## order p * q ^ 2
##
ID_GROUP_FUNCS[ 6 ] := function( G, inforec )
local n, p, q, s1, s2, typ, lat, nor, non, x, s, A, B, C, AC, BC, PO,id;
# get primes
n := Size(G);
p := FactorsInt(n);
q := p[3];
p := p[1];
# compute the sylow subgroups
s1 := SylowSubgroup( G, q );
s2 := SylowSubgroup( G, p );
if IsAbelian( G ) then
if IsCyclic( s1 ) then
typ := "pq2";
else
typ := "pqq";
fi;
elif ( p <> 2 ) and ( (q+1) mod p = 0 ) then
typ := "Npq2";
elif IsCyclic( s1 ) then
typ := "Mpq2";
elif not IsTrivial( Centralizer( s1, s2 ) ) then
typ := "Dpqxq";
else
lat := List( ConjugacyClassesSubgroups(s1), Representative );
lat := Filtered( lat, t -> Size(t) = q );
nor := [];
non := [];
x := 1;
while x <= Length(lat) and 0 = Length(non) and Length(nor) < 3 do
if IsNormal( G, lat[x] ) then
Add( nor, lat[x] );
else
Add( non, lat[x] );
fi;
x := x + 1;
od;
if 0 = Length(non) and 2 < Length(nor) then
# Kpq2;
typ := 1;
else
# Lpq2
while x <= Length(lat) and Length(nor) < 2 do
if IsNormal( G, lat[x] ) then
Add( nor, lat[x] );
fi;
x := x + 1;
od;
A := GeneratorsOfGroup( nor[1] )[1];
B := GeneratorsOfGroup( nor[2] )[1];
C := GeneratorsOfGroup( s2 )[1];
AC := A^C;
x := 1;
PO := 0;
while PO <> AC do
x := x+1;
if x^p mod q = 1 then
PO := A^x;
fi;
od;
BC := B^C;
s := 1;
PO := 0;
while s < q and PO <> BC do
s := s+1;
if s mod p <> 0 and s mod p <> 1 then
PO := B^((x^s) mod q);
fi;
od;
s := s mod p;
if ((1/s) mod p) < s then s := (1/s) mod p; fi;
typ := s;
fi;
fi;
return Position(
NUMBER_SMALL_GROUPS_FUNCS[ 6 ]( Size( G ), inforec ).types, typ );
end;
#############################################################################
##
#F ID_GROUP_FUNCS[ 7 ]( G, inforec )
##
## order p * q * r
##
ID_GROUP_FUNCS[ 7 ] := function( G, inforec )
local n, p, q, r, s1, s2, s3, typ, PO, A, C, AC, x, s, y, B, BC, id;
# get primes
n := Size(G);
p := FactorsInt(n);
r := p[3];
q := p[2];
p := p[1];
# compute the sylow subgroups
s1 := SylowSubgroup( G, r );
s2 := SylowSubgroup( G, q );
s3 := SylowSubgroup( G, p );
if IsAbelian( G ) then
typ := "pqr";
elif IsNormal( G, s3 ) then
typ := "Dqrxp";
elif not IsAbelian( ClosureGroup( s1, s2 ) ) then
typ := "Hpqr";
elif IsAbelian( ClosureGroup( s1, s3 ) ) then
typ := "Dpqxr";
elif IsAbelian( ClosureGroup( s2, s3 ) ) then
typ := "Dprxq";
else
# find <A> and <C>
C := GeneratorsOfGroup( SylowSubgroup(G,p) )[1];
A := GeneratorsOfGroup( SylowSubgroup(G,r) )[1];
AC := A^C;
PO := 0;
s := 1;
while AC <> PO do
s := s+1;
if s mod r <> 1 and s^p mod r = 1 then
PO := A^s;
fi;
od;
# correct <C>
x := First( [2..r-1], t -> t^p mod r = 1 );
s := LogMod( x, s, r );
C := C^s;
# now find <B>
B := GeneratorsOfGroup( SylowSubgroup(G,q) )[1];
BC := B^C;
PO := 0;
s := 1;
while BC <> PO do
s := s+1;
if s mod r <> 1 and s^p mod r = 1 then
PO := B^s;
fi;
od;
# and find <s>
y := First( [2..q-1], t -> t^p mod q = 1 );
s := LogMod( s, y, q ) mod OrderMod( y, q );
typ := s; # the typ is Gpqr( s )
fi;
return Position(
NUMBER_SMALL_GROUPS_FUNCS[ 7 ]( Size( G ), inforec ).types, typ );
end;
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