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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346#############################################################################
##
#W smlgp9.g GAP group library Hans Ulrich Besche
## Bettina Eick, Eamonn O'Brien
## Mike Newman, Boris Girnat
##
## This file contains the generic construction of groups of order p^4, p^5
## and p^6 with order > 3125 (groups with order up to 3125 are contained in
## the lower layers of the small groups library.
##
#############################################################################
##
## tell GAP about the component
##
DeclareComponent("small9","1.0");
#############################################################################
##
#F SMALL_AVAILABLE_FUNCS[ 9 ]
##
SMALL_AVAILABLE_FUNCS[ 9 ] := function( size )
local p;
if ( not IsPrimePowerInt( size ) ) or size <= 3125 then
return fail;
fi;
p := FactorsInt( size );
if Length( p ) = 4 then
return rec( func := 19,
number := 15,
lib := 9,
p := p[ 1 ] );
fi;
if Length( p ) = 5 then
return rec( func := 20,
number := 61 +
2 * p[ 1 ] +
2 * Gcd( 3, p[ 1 ] - 1 ) +
Gcd( 4, p[ 1 ] - 1 ),
lib := 9,
p := p[ 1 ] );
fi;
if Length( p ) = 6 then
return rec( func := 21,
lib := 9,
p := p[ 1 ] );
fi;
return fail;
end;
#############################################################################
##
#F SMALL_GROUP_FUNCS[ 19 ]( size, i, inforec )
##
## order p^4 , p >= 11
##
SMALL_GROUP_FUNCS[ 19 ] := function( size, i, inforec )
local g, f, p, c, w;
if i > 15 then
Error( "there are just 15 groups of size ", size );
fi;
f := FreeGroup( 4 );
p := inforec.p;
c := CombinatorialCollector( f, [p,p,p,p] );
w := IntFFE( Z ( p ) );
if i = 1 then
SetPower(c,1,f.2);
SetPower(c,2,f.3);
SetPower(c,3,f.4);
elif i = 2 then
SetPower(c,1,f.3);
SetPower(c,2,f.4);
elif i = 3 then
SetCommutator(c,2,1,f.3);
SetPower(c,1,f.4);
elif i = 4 then
SetCommutator(c,2,1,f.3);
SetPower(c,1,f.4);
SetPower(c,2,f.3);
elif i = 5 then
SetPower(c,1,f.3);
SetPower(c,3,f.4);
elif i = 6 then
SetCommutator(c,2,1,f.4);
SetPower(c,1,f.3);
SetPower(c,3,f.4);
elif i = 7 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
elif i = 8 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetPower(c,1,f.4);
elif i = 9 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetPower(c,2,f.4);
elif i = 10 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetPower(c,2,f.4^w);
elif i = 11 then
SetPower(c,1,f.4);
elif i = 12 then
SetCommutator(c,2,1,f.4);
elif i = 13 then
SetCommutator(c,2,1,f.4);
SetPower(c,1,f.4);
elif i = 14 then
SetCommutator(c,2,1,f.4);
SetPower(c,3,f.4);
elif i = 15 then
# elementary abelian group
fi;
g := GroupByRwsNC( c );
SetIsPGroup( g, true );
return g;
end;
#############################################################################
##
#F SMALL_GROUP_FUNCS[ 20 ]( size, i, inforec )
##
## order p^5 , p >= 7
##
SMALL_GROUP_FUNCS[ 20 ] := function( size, i, inforec )
local g, typ, k, f, c, w, p, a, b;
if i > inforec.number then
Error( "there are just ", inforec.number, " groups of size ", size );
fi;
f := FreeGroup( 5 );
p := inforec.p;
c := CombinatorialCollector( f, [p,p,p,p,p] );
w := IntFFE( Z ( p ) );
a := Gcd( p-1, 3 );
b := Gcd( p-1, 4 );
if i <= 10 then
typ := i;
elif i <= 11+(p-1)/2-1 then
typ := 11;
k := i - 10;
elif i <= 11+p-2 then
typ := 12;
k := i - (11+(p-1)/2-1);
elif i <= 25+p then
typ := i - p + 3;
elif i <= 25+p+a then
typ := 29;
k := i - (25+p);
elif i <= 27+p+a then
typ := i - p - a + 4;
elif i <= 27+p+2*a then
typ := 32;
k := i - (27+p+a);
elif i <= 27+p+2*a+b then
typ := 33;
k := i - (27+p+2*a);
elif i <= 41+p+2*a+b then
typ := i - p - 2*a - b + 6;
elif i <= 41+p+2*a+b+(p-1)/2 then
typ := 48;
k := i - (41+p+2*a+b);
elif i = 42+p+2*a+b+(p-1)/2 then
typ := 49;
elif i <= 42+p+2*a+b+p-1 then
typ := 50;
k := i - (42+p+2*a+b+(p-1)/2);
else
typ := i - 2*p - 2*a - b + 9;
fi;
if typ = 1 then
SetPower(c,1,f.2);
SetPower(c,2,f.3);
SetPower(c,3,f.4);
SetPower(c,4,f.5);
elif typ = 2 then
SetCommutator(c,2,1,f.3);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 3 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
elif typ = 4 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.5);
elif typ = 5 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.4);
elif typ = 6 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.4^w);
elif typ = 7 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 8 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4*f.5);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 9 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4*f.5^w);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 10 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5^w);
SetCommutator(c,3,2,f.4);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 11 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5^(w^k mod p));
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 12 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4*f.5^(w^k mod p));
SetCommutator(c,3,2,f.4^(w^(k-1) mod p)*f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 13 then
SetPower(c,1,f.3);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 14 then
SetCommutator(c,2,1,f.5);
SetPower(c,1,f.3);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 15 then
SetCommutator(c,2,1,f.3);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 16 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetPower(c,1,f.4);
elif typ = 17 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 18 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5^w);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 19 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 20 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
elif typ = 21 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,2,f.5);
elif typ = 22 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 23 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5^w);
SetCommutator(c,3,2,f.5^w);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 24 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.5);
SetCommutator(c,4,2,f.5^(p-1));
SetPower(c,1,f.4);
SetPower(c,2,f.3);
SetPower(c,3,f.5);
elif typ = 25 then
SetCommutator(c,2,1,f.3);
SetPower(c,1,f.4);
SetPower(c,2,f.3);
SetPower(c,4,f.5);
elif typ = 26 then
SetPower(c,1,f.3);
SetPower(c,3,f.4);
SetPower(c,4,f.5);
elif typ = 27 then
SetCommutator(c,2,1,f.5);
SetPower(c,1,f.3);
SetPower(c,3,f.4);
SetPower(c,4,f.5);
elif typ = 28 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5);
elif typ = 29 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5^(w^k mod p));
SetPower(c,2,f.5);
elif typ = 30 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5);
SetPower(c,1,f.5);
elif typ = 31 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5);
SetCommutator(c,3,2,f.5);
elif typ = 32 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5^(w^k mod p));
SetCommutator(c,3,2,f.5^(w^k mod p));
SetPower(c,2,f.5);
elif typ = 33 then
SetCommutator(c,2,1,f.3);
SetCommutator(c,3,1,f.4);
SetCommutator(c,4,1,f.5^(w^k mod p));
SetCommutator(c,3,2,f.5^(w^k mod p));
SetPower(c,1,f.5);
elif typ = 34 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,1,f.5);
elif typ = 35 then
SetCommutator(c,2,1,f.4);
SetPower(c,3,f.5);
elif typ = 36 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,3,f.5);
elif typ = 37 then
SetCommutator(c,3,2,f.4);
SetPower(c,3,f.5);
elif typ = 38 then
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,3,f.5);
elif typ = 39 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,3,f.4);
elif typ = 40 then
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 41 then
SetCommutator(c,3,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 42 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,2,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 43 then
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 44 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,1,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 45 then
SetCommutator(c,2,1,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 46 then
SetCommutator(c,2,1,f.4*f.5);
SetCommutator(c,3,1,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 47 then
SetCommutator(c,3,1,f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 48 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,3,1,f.5^(w^k mod p));
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 49 then
SetCommutator(c,2,1,f.5^w);
SetCommutator(c,3,1,f.4);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 50 then
SetCommutator(c,2,1,f.4*f.5^(w^k mod p));
SetCommutator(c,3,1,f.4^(w^(k-1) mod p)*f.5);
SetPower(c,2,f.4);
SetPower(c,3,f.5);
elif typ = 51 then
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 52 then
SetCommutator(c,3,2,f.5);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 53 then
SetCommutator(c,3,1,f.5);
SetPower(c,1,f.4);
SetPower(c,4,f.5);
elif typ = 54 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
elif typ = 55 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetPower(c,3,f.5);
elif typ = 56 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetPower(c,2,f.5);
elif typ = 57 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetPower(c,1,f.5);
elif typ = 58 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5^w);
SetPower(c,1,f.5);
elif typ = 59 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetCommutator(c,3,1,f.5);
elif typ = 60 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetCommutator(c,3,1,f.5);
SetPower(c,3,f.5);
elif typ = 61 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetCommutator(c,3,1,f.5);
SetPower(c,2,f.5);
elif typ = 62 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5);
SetCommutator(c,3,1,f.5);
SetPower(c,1,f.5);
elif typ = 63 then
SetCommutator(c,2,1,f.4);
SetCommutator(c,4,2,f.5^w);
SetCommutator(c,3,1,f.5^w);
SetPower(c,1,f.5);
elif typ = 64 then
SetCommutator(c,2,1,f.5);
elif typ = 65 then
SetCommutator(c,2,1,f.5);
SetCommutator(c,4,3,f.5);
elif typ = 66 then
SetPower(c,1,f.5);
elif typ = 67 then
SetCommutator(c,2,1,f.5);
SetPower(c,1,f.5);
elif typ = 68 then
SetCommutator(c,2,1,f.5);
SetPower(c,3,f.5);
elif typ = 69 then
SetCommutator(c,2,1,f.5);
SetCommutator(c,4,3,f.5);
SetPower(c,4,f.5);
elif typ = 70 then
# elementary abelian group
fi;
g := GroupByRwsNC( c );
SetIsPGroup( g, true );
return g;
end;
#############################################################################
##
#F SMALL_GROUP_FUNCS[ 21 ]( size, i, inforec )
##
## order p^6 , p >= 5
##
SMALL_GROUP_FUNCS[ 21 ] := function( size, i, inforec )
local n, p, phi, part,
j, ind, ri, g, j1, j2, k, l, m, c1, c2,
r, mem, rel,
F, famRels, grpRels;
if not IsBound( inforec.F ) then
inforec := NUMBER_SMALL_GROUPS_FUNCS[ 21 ]( size, inforec );
fi;
n := inforec.number;
if i > n then
Error( "there are just ", n, " groups of size ", size );
fi;
if i <= 11 then
part := [ [ 6 ],
[ 5, 1 ],
[ 4, 2 ], [ 4, 1, 1 ],
[ 3, 3 ], [ 3, 2, 1 ], [ 3, 1, 1, 1 ],
[ 2, 2, 2 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ],
[ 1, 1, 1, 1, 1, 1 ] ];
return AbelianGroup( IsPcGroup, List( part[i], x -> inforec.p^x ) );
fi;
F := inforec.F;
p := inforec.p;
phi := 0;
repeat
phi := phi + 1;
i := i - inforec.num[ phi ];
until i <= 0;
i := i + inforec.num[ phi ];
if not IsBound( SMALL_GROUP_LIB[ 1 ] ) then
ReadSmallLib( "sml", 9, 1, [ ] );
SMALL_GROUP_LIB[ 1 ][ 1 ] := inforec;
fi;
if not IsBound( inforec.groups[ phi ] ) then
inforec.groups[ phi ] := [];
for j in [ 1 .. Length( SMALL_GROUP_LIB[ 1 ][ phi ] ) ] do
n := SMALL_GROUP_LIB[ 1 ][ phi ][ j ][ 1 ];
r := rec( classBound := n mod 7,
famMems := [ ] );
n := QuoInt( n, 7 );
famRels := ShallowCopy( inforec.genRels );
while n > 0 do
Unbind( famRels[ n mod 22 ] );
n := QuoInt( n, 22 );
od;
n := SMALL_GROUP_LIB[ 1 ][ phi ][ j ][ 2 ];
while n > 0 do
ind := n mod 22;
n := QuoInt( n, 22 );
ri := One( F );
l := n mod 3;
n := QuoInt( n, 3 );
for m in [ 1 .. l ] do
g := F.(n mod 7);
n := QuoInt( n, 7 );
j2 := n mod 50;
n := QuoInt( n, 50 );
ri := ri * g^SMALL_GROUP_FUNCS[23]( j2, inforec, fail );
od;
famRels[ ind ] := famRels[ ind ] / ri;
od;
r.famRels := famRels;
for n in SMALL_GROUP_LIB[ 1 ][ phi ][ j ][ 3 ] do
c1 := n mod 9;
n := QuoInt( n, 9 );
c2 := n mod 22;
n := QuoInt( n, 22 );
mem := rec( num := SMALL_GROUP_FUNCS[ 22 ]( c1, c2, p ),
rels := [] );
if mem.num > 0 then
while n > 0 do
rel := rec( ind := n mod 22,
exli := [ ] );
n := QuoInt( n, 22 );
l := n mod 3;
n := QuoInt( n, 3 );
for m in [ 1 .. l ] do
j1 := n mod 7;
n := QuoInt( n, 7 );
j2 := n mod 50;
n := QuoInt( n, 50 );
Add( rel.exli, [ j1, j2 ] );
od;
Add( mem.rels, rel );
od;
Add( r.famMems, mem );
fi;
od;
Add( inforec.groups[ phi ], r );
od;
fi;
j := 1;
k := 1;
while i > inforec.groups[ phi ][ j ].famMems[ k ].num do
i := i - inforec.groups[ phi ][ j ].famMems[ k ].num;
k := k + 1;
if k > Length( inforec.groups[ phi ][ j ].famMems ) then
j := j + 1;
k := 1;
fi;
od;
grpRels := ShallowCopy( inforec.groups[ phi ][ j ].famRels );
for rel in inforec.groups[ phi ][ j ].famMems[ k ].rels do
ri := One( F );
for m in rel.exli do
ri := ri * F.(m[1])^SMALL_GROUP_FUNCS[23]( m[2], inforec, i );
od;
grpRels[ rel.ind ] := grpRels[ rel.ind ] / ri;
od;
## return Image( EpimorphismQuotientSystem( PQuotient( F/Set( grpRels ),
## p, inforec.groups[ phi ][ j ].classBound ) ) );
j:= EpimorphismQuotientSystem( PQuotient( F/Set( grpRels ),
p, inforec.groups[ phi ][ j ].classBound ) );
i := Image( j );
i!.eqs := j;
return i;
end;
#############################################################################
##
#F SMALL_GROUP_FUNCS[ 22 ]( c1, c2, p )
##
##
SMALL_GROUP_FUNCS[ 22 ] := function( c1, c2, p )
if c1 = 0 or
c1 = 1 and p mod 3 = 1 or
c1 = 2 and p mod 4 = 1 or
c1 = 3 and p mod 4 = 3 or
c1 = 4 and p mod 5 = 1 or
c1 = 5 and p = 5 or
c1 = 6 and p <> 5 or
c1 = 7 and p <> 5 and p mod 4 = 1 or
c1 = 8 and p <> 5 and p mod 3 = 2 then
if c2 <= 5 then
return c2;
elif c2 = 6 then
return p;
elif c2 = 7 then
return p-1;
elif c2 = 8 then
return p-2;
elif c2 = 9 then
return p-3;
elif c2 = 10 then
return p - 3 + Gcd( p-1, 4 ) / 2;
elif c2 = 11 then
return (p-1)/2;
elif c2 = 12 then
return (p-3)/2;
elif c2 = 13 then
return 2*p-2;
elif c2 = 14 then
return 2*p-4;
elif c2 = 15 then
return p*(p-1)/2;
elif c2 = 16 then
return (p-1)*(p-1)/2;
elif c2 = 17 then
return p*(p-1)/2 - 11/4*p + (1/4*p-1)*Gcd(p-1,4) + 23/4;
elif c2 = 18 then
return p*(p-1)/2 + 3/4*p + (-1/4*p+1/2)*Gcd(p-1,4) - 7/4;
elif c2 = 19 then
return (p-2)*(p-1)/2;
elif c2 = 20 then
return (p-2)*(p-1)/2-1;
elif c2 = 21 then
return (p-2)*(p-1)/2+1;
fi;
else
return 0;
fi;
end;
#############################################################################
##
#F SMALL_GROUP_FUNCS[ 23 ]( j2, inforec )
##
##
SMALL_GROUP_FUNCS[ 23 ] := function( j2, inforec, ii )
local p, pr, nqr, nqrm, squares,
cart, f, gg, k, r, s, t, x, y, z;
p := inforec.p;
pr := inforec.pr;
nqr := inforec.nqr;
nqrm := inforec.nqrm;
squares := inforec.squares;
if j2 in [ 20, 21, 45, 46 ] and not IsBound( inforec.epx ) then
x := 1;
y := 0;
repeat
y := y + 1;
if y = p then
x := x + 1;
y := 1;
fi;
until ( x*x - nqrm*y*y ) mod p = nqrm;
inforec.epx := x;
inforec.epy := y;
fi;
if j2 = 0 then
return -1;
elif j2 <= 2 then
return j2;
elif j2 <= 7 then
return j2 + ii - 5;
elif j2 = 8 then
return -ii + 1;
elif j2 = 9 then
if ii = 1 then
return 0;
fi;
return -ii;
elif j2 <= 21 then
if inforec.activeCache <> j2 then
inforec.activeCache := j2;
if j2 = 10 then
inforec.listCache :=
Filtered( Cartesian( [ 1, nqr ], [ 1 .. p-1 ] ),
x -> ( 1 + 4*x[1]*x[2] ) mod p <> 0 and
( 1 + 4*x[1]*x[2] ) mod p in squares );
elif j2 = 11 then
inforec.subCache := Cartesian( [ 1, nqr ], [ 1 .. p-1 ] );
inforec.subCacheLength := [ ];
for r in [ 1, nqr ] do
for s in [ 1 .. p-1 ] do
x := 0;
for k in List( [ 0 .. (p-1)/2 ], x -> pr^x mod p ) do
if k <> r*s mod p and
((1-k)^2+4*r*s) mod p <> 0 and
((1-k)^2+4*r*s) mod p in squares and
not ( r=nqr and
k in [1,p-1] and
( -s mod p in squares ) ) then
x := x + 1;
fi;
od;
Add( inforec.subCacheLength, x );
od;
od;
elif j2 = 12 then
inforec.listCache :=
Filtered( Cartesian( [ 1, nqr ], [ 1 .. p-1 ] ),
x -> not(
( 1 + 4*x[1]*x[2] ) mod p in squares ) );
elif j2 = 13 then
inforec.subCache := Cartesian( [ 1, nqr ], [ 1 .. p-1 ] );
inforec.subCacheLength := [ ];
for r in [ 1, nqr ] do
for s in [ 1 .. p-1 ] do
x := 0;
for k in List( [ 0 .. (p-1)/2 ], x -> pr^x mod p ) do
if k <> r*s mod p and
not ( ( (1-k)^2+4*r*s ) mod p ) in squares and
not ( r=nqr and
k in [1,p-1] and
( -s mod p in squares ) ) then
x := x + 1;
fi;
od;
Add( inforec.subCacheLength, x );
od;
od;
elif j2 = 14 then
inforec.listCache :=
Filtered( Cartesian( [ 1, nqr ], [ 0 .. p-1 ] ),
x -> ( 1 + 4*x[1]*x[2] ) mod p = 0 );
elif j2 = 15 then
inforec.listCache := [];
for r in [ 1, nqr ] do
for k in List( [ 1 .. (p-3)/2 ], x -> pr^x mod p ) do
for t in [ 0 .. p-1 ] do
if ( 4*r*t + (1-k)^2) mod p = 0 then
Add( inforec.listCache, [ r, t, k ] );
fi;
od;
od;
od;
elif j2 = 16 then
x := Int( p / 4 );
inforec.subCache := [ 1 .. p-1 ];
inforec.subCacheLength :=
Concatenation( [ 1 .. x ] * 0 + (p-3)/2,
[ 1 .. p - 1 - 2*x ] * 0 + (p-1)/2,
[ 1 .. x ] * 0 + (p+1)/2 );
elif j2 = 17 then
inforec.listCache := [];
for r in [ 1, nqr ] do
for gg in List( [ 0 .. (p-3)/2 ], x -> pr^x mod p ) do
if not ( p mod 4 = 3 and
r = nqr and
gg = 1 ) then
Add( inforec.listCache, [r,gg/r mod p,gg] );
fi;
od;
od;
elif j2 = 18 then
inforec.listCache := List(
Filtered( [ 1 .. p-1 ], x-> x^2 mod p <> p-nqr ),
x -> [ nqrm * x mod p, x ] );
elif j2 = 19 then
if p mod 4 = 1 then
inforec.subCache := [ 1 .. p-1 ];
inforec.subCacheLength := [ 1 .. p-1 ] * 0 + ((p-1)/2);
for x in Filtered( [ 2 .. p-2 ],
z -> (z^2-1)/nqrm mod p in squares ) do
inforec.subCacheLength[ x ] := (p-3)/2;
od;
else
inforec.subCache := [ 1 .. (p-1)/2 ];
inforec.subCacheLength := [ 1 .. (p-1)/2 ] * 0 + (p-1);
for x in Filtered( [ 2 .. (p-1)/2 ],
z -> (z^2-1)/nqrm mod p in squares ) do
inforec.subCacheLength[ x ] := p - 3;
od;
fi;
elif j2 = 20 then
if p mod 4 = 1 then
cart := Filtered( [ 1..p-1 ], l -> l^2 mod p <> p-1 );
else
cart := [ 1 .. (p-1)/2 ];
fi;
inforec.listCache := List( cart,
c -> [ nqrm * ( c - inforec.epy ) mod p , -inforec.epx,
inforec.epx, ( c + inforec.epy ) mod p] );
elif j2 = 21 then
if p mod 4 = 1 then
inforec.subCache := [ 1 .. (p-1)/2 ];
inforec.subCacheLength := [ 1 .. (p-1)/2 ] * 0 + (p-1);
for x in Filtered( [ 1 .. (p-1)/2 ],
z -> (z^2*nqr-1) mod p in squares ) do
inforec.subCacheLength[ x ] := p - 3;
od;
else
inforec.subCache := [ 1 .. p-1 ];
inforec.subCacheLength := [ 1 .. p-1 ] * 0 + ((p-1)/2);
for x in Filtered( [ 1 .. p-1 ],
z -> (z^2*nqr-1) mod p in squares ) do
inforec.subCacheLength[ x ] := (p-3)/2;
od;
fi;
fi;
inforec.activeSubCache := fail;
fi;
if j2 in [ 11, 13, 16, 19, 21 ] then
x := 1;
while ii > inforec.subCacheLength[ x ] do
ii := ii - inforec.subCacheLength[ x ];
x := x + 1;
od;
if x <> inforec.activeSubCache then
inforec.activeSubCache := x;
x := inforec.subCache[ x ];
inforec.listCache := [];
if j2 = 11 then
r := x[ 1 ];
s := x[ 2 ];
for k in List( [ 0 .. (p-1)/2 ], x -> pr^x mod p ) do
if k <> r*s mod p and
((1-k)^2+4*r*s) mod p <> 0 and
((1-k)^2+4*r*s) mod p in squares and
not ( r=nqr and
k in [1,p-1] and
( -s mod p in squares ) ) then
Add( inforec.listCache, [ r, s, k ] );
fi;
od;
elif j2 = 13 then
r := x[ 1 ];
s := x[ 2 ];
for k in List( [ 0 .. (p-1)/2 ], x -> pr^x mod p ) do
if k <> r*s mod p and
not ( ( (1-k)^2+4*r*s ) mod p ) in squares and
not ( r=nqr and
k in [1,p-1] and
( -s mod p in squares ) ) then
Add( inforec.listCache, [ r, s, k ] );
fi;
od;
elif j2 = 16 then
for z in Filtered( [ 0 .. (p-1)/2 ],
y -> x <> y and 2*x mod p <> y ) do
Add( inforec.listCache, [ x, z - x ] );
od;
elif j2 = 19 then
if p mod 4 = 1 then
k := (p-1) / 2;
else
k := p - 1;
fi;
for z in Filtered( [ 1 .. k ],
c -> ( x^2 - nqrm*c^2 ) mod p <> 1 ) do
Add( inforec.listCache, [ nqrm * z mod p, x - 1,
x + 1, z ] );
od;
elif j2 = 21 then
if p mod 4 = 1 then
k := p - 1;
else
k := (p-1) / 2;
fi;
for z in Filtered( [ 1 .. k ],
c -> ( x^2 - nqrm*c^2 ) mod p <> nqrm ) do
Add( inforec.listCache,
[ nqrm * ( z - inforec.epy ) mod p,
x - inforec.epx,
( x + inforec.epx ) mod p,
( z + inforec.epy ) mod p ] );
od;
fi;
fi;
fi;
inforec.ExpCache := inforec.listCache[ ii ];
return inforec.ExpCache[ 1 ];
elif j2 <= 24 then
return inforec.ExpCache[ j2 - 20 ];
elif j2 = 25 then
return nqr;
elif j2 = 26 then
return nqrm;
elif j2 = 27 then
return -nqr;
elif j2 = 28 then
return -nqrm;
elif j2 = 29 then
return 2*nqr;
elif j2 = 30 then
if ii mod 2 = 1 then
return 1;
else
return nqr;
fi;
elif j2 = 31 then
if ii <= 2 then
return 1;
else
return nqr;
fi;
elif j2 = 32 then
if ii = 1 then
return -1;
else
return -nqrm;
fi;
elif j2 = 33 then
return nqr * ii;
elif j2 = 34 then
return nqrm * ii;
elif j2 = 35 then
return pr^ii mod p;
elif j2 = 36 then
return 2*pr^ii mod p;
elif j2 = 37 then
return -pr^ii mod p;
elif j2 = 38 then
return ( -1 + pr^(ii*2-1) ) mod p;
elif j2 = 39 then
x := 1;
y := 0;
repeat
y := y + 1;
if y = p then
x := x + 1;
y := 1;
fi;
until ( x*x - nqrm*y*y ) mod p = ii+1;
inforec.ExpCache := [ , -nqrm*y, y, 1+x ];
return 1-x;
elif j2 = 40 then
if ii < p then
if ii = 1 then
inforec.ExpCache := [ , 0 ];
else
inforec.ExpCache := [ , ii ];
fi;
return 1;
else
if ii < p + nqrm then
inforec.ExpCache := [ , ii - p ];
else
inforec.ExpCache := [ , ii - p + 1 ];
fi;
return nqr;
fi;
elif j2 = 41 then
if ii < p then
inforec.ExpCache := [ , ii ];
return 1;
else
inforec.ExpCache := [ , ii - ( p-1 ) ];
return nqr;
fi;
elif j2 = 42 then
if ii <= p-2 then
inforec.ExpCache := [ , ii + 1 ];
return 1;
else
inforec.ExpCache := [ , ii - p + 3 ];
return nqr;
fi;
elif j2 = 43 then
if ii <= (p-1)/2 then
inforec.ExpCache := [ , ii ];
return 1;
else
inforec.ExpCache := [ , ii - ( p-1 ) / 2 ];
return nqr;
fi;
elif j2 = 44 then
x := Int( (ii-1) / ((p-1)/2) );
inforec.ExpCache := [ , ii - x * (p-1)/2 ];
return x;
elif j2 = 45 then
inforec.ExpCache := [ , -inforec.epx,
inforec.epx, inforec.epy ];
return -nqrm * inforec.epy mod p;
elif j2 = 46 then
inforec.ExpCache := [ , ii - inforec.epx,
( ii + inforec.epx ) mod p, inforec.epy ];
return -nqrm * inforec.epy mod p;
elif j2 = 47 then
return -( (ii-2) / 3 mod 5 );
elif j2 <= 49 then
if j2 = 48 then
f := 1;
else
f := nqrm;
fi;
x := 0;
y := 0;
repeat
y := y + 1;
if y = p then
x := x + 1;
y := 0;
fi;
until ( x*x - f*y*y ) mod p = ii;
inforec.ExpCache := [ , -y+1 ];
return -x-1;
fi;
end;
#############################################################################
##
#F SELECT_SMALL_GROUPS_FUNCS[ 19, 20, 21 ]( funcs, vals, inforec, all, id )
##
SELECT_SMALL_GROUPS_FUNCS[ 19 ] := SELECT_SMALL_GROUPS_FUNCS[ 11 ];
SELECT_SMALL_GROUPS_FUNCS[ 20 ] := SELECT_SMALL_GROUPS_FUNCS[ 11 ];
SELECT_SMALL_GROUPS_FUNCS[ 21 ] := SELECT_SMALL_GROUPS_FUNCS[ 11 ];
#############################################################################
##
#F NUMBER_SMALL_GROUPS_FUNCS[ 21 ]( size, inforec )
##
## order p^6 , p >= 6
##
NUMBER_SMALL_GROUPS_FUNCS[ 21 ] := function( size, inforec )
local p, p2, pp2, a, b, c, i, j;
if IsBound( SMALL_GROUP_LIB[ 1 ] ) and
IsBound( SMALL_GROUP_LIB[ 1 ][ 1 ] ) and
SMALL_GROUP_LIB[ 1 ][ 1 ].p = inforec.p then
return SMALL_GROUP_LIB[ 1 ][ 1 ];
fi;
p := inforec.p;
p2 := (p-1) / 2;
pp2 := p*p2;
a := Gcd( 3, p-1 );
b := Gcd( 4, p-1 );
c := Gcd( 5, p-1 );
inforec.num := [ 11, 31, 32, 3*p+32, 7, 2*p+21, 21, p+5, 3*a+7,
3*a+3*b+4, 2*p+10, p+13, p+10, 3, p+3, p+a+12, 4*p+a+30, 3*p+a+b+9,
3*pp2+6*p+p2+11, 5*p+a+b+13, 3*pp2+4*p-p2+2, 7, p+4*a+b+5, a+3, p2+2,
p2+2, a+b+3, p, p, 2*a+4, 7, 5, 6, 3, b+2, 2*a+b+1, b+4, p+b+c,
p+2*a+c, a+2, a+1, p+1, p];
inforec.number := Sum( inforec.num );
inforec.F := FreeGroup( IsSyllableWordsFamily, 6, "q" );
inforec.genRels := [];
for i in [ 1 .. 6 ] do
Add( inforec.genRels, inforec.F.(i)^p );
od;
for j in [ 2 .. 6 ] do
for i in [ 1 .. j-1 ] do
Add( inforec.genRels, Comm( inforec.F.(i), inforec.F.(j) ) );
od;
od;
inforec.groups := [];
inforec.activeCache := fail;
inforec.pr := Int( PrimitiveRoot( GF( p ) ) );
inforec.squares := Set( List( [ 0 .. p - 1 ], x -> x ^ 2 mod p ) );
inforec.nqr := 1;
repeat
inforec.nqr := inforec.nqr + 1;
until not inforec.nqr in inforec.squares;
inforec.nqrm := 1 / inforec.nqr mod p;
if IsBound( SMALL_GROUP_LIB[ 1 ] ) then
SMALL_GROUP_LIB[ 1 ][ 1 ] := inforec;
fi;
return inforec;
end;