Constructing Co3 in GAP 3

See also the corresponding GAP 4 page.

The original example page provides links to a GAP 3 version and to an early GAP 4 version. The present page contains just a GAP 3 version. The associated GAP 3 input file is also available.

  gap> x:=Indeterminate(GF(2));
  X(GF(2))
  gap> x.name:="x";;
  gap> f:=x^23-1;
  Z(2)^0*(x^23 + 1)
  gap> Factors(f);
  [ Z(2)^0*(x + 1), Z(2)^0*(x^11 + x^10 + x^6 + x^5 + x^4 + x^2 + 1), 
    Z(2)^0*(x^11 + x^9 + x^7 + x^6 + x^5 + x + 1) ]
  gap> f:=First(Factors(f),i->Degree(i)>1);
  Z(2)^0*(x^11 + x^10 + x^6 + x^5 + x^4 + x^2 + 1)
  gap> RequirePackage("guava");
  
                ___________                   |                  
               /            \           /   --+--  Version 1.3   
              /      |    | |\\        //|    |                  
             |    _  |    | | \\      // |                        
             |     \ |    | |--\\    //--|     Jasper Cramwinckel  
              \     ||    | |   \\  //   |     Erik Roijackers     
               \___/  \___/ |    \\//    |     Reinald Baart       
                                               Eric Minkes       
  gap> cod:=GeneratorPolCode(f,23,GF(2));
  a cyclic [23,12,1..7]3 code defined by generator polynomial over GF(2)
  gap> IsPerfectCode(cod);
  true
  gap> ext:=ExtendedCode(cod);
  a linear [24,12,8]4 extended code
  gap> WeightDistribution(ext);
  [ 1, 0, 0, 0, 0, 0, 0, 0, 759, 0, 0, 0, 2576, 0, 0, 0, 759, 0, 0, 0, 0, 0, 0, 
    0, 1 ]
  gap> m24:=AutomorphismGroup(ext); m24.name:="m24";
  Group( ( 1, 2)( 6,15,24,18)( 7,14,12,19)( 9,17,16,20)(10,21,23,22)(11,13), 
  ( 2, 3)( 6,10)( 7,24)( 9,17)(11,13)(12,18)(14,21)(15,23), ( 3, 4)( 7,19)
  (10,22)(11,13)(12,14)(15,18)(17,20)(21,23), ( 4, 5)( 6,23,21,16)( 7, 9,20,10)
  (11,13)(12,15,17,19)(14,22,18,24), ( 5, 9,11,17)( 6,19,24,21)( 7,23)
  ( 8,20,13,16)(10,12)(14,15,22,18), ( 5, 8)( 7,23)( 9,16)(10,12)(11,13)(14,22)
  (17,20)(19,21), ( 6,10,19)( 7,14,18)( 8,11,13)( 9,17,16)(12,21,24)(15,23,22), 
  ( 6,19)( 7, 9)(10,20)(12,16)(14,24)(15,21)(17,23)(18,22), ( 6, 7)( 9,19)
  (10,15)(12,24)(14,16)(17,22)(18,23)(20,21), ( 6,18)( 7,23)( 9,17)(10,12)
  (14,21)(15,24)(16,20)(19,22), ( 6,15)( 7,10)( 9,20)(12,23)(14,22)(16,17)
  (18,24)(19,21) )
  "m24"
  gap> Size(m24);
  244823040
  gap> Transitivity(m24,[1..24]);
  5
  gap> m22a:=Stabilizer(m24,[23,24],OnSets);
  Subgroup( m24, [ ( 1, 2)( 5,14)( 6,18)( 7,12)( 8,19)( 9,20)(11,21)(13,22), 
    ( 1, 3)( 4,13, 7,21)( 5,14)( 6, 9,22,11)( 8,16,15,19)(10,18,12,20), 
    ( 4,13)( 6,19)( 7,21)( 8,11)( 9,15)(10,20)(12,18)(16,22), 
    ( 3, 4,14,22)( 5,16,17,13)( 6, 7)( 8,15,11,10)( 9,12,20,18)(19,21), 
    ( 3,17)( 4,16)( 5,14)( 6, 7)( 8,11)(12,18)(13,22)(19,21), 
    ( 4,22,16)( 5,17,14)( 6, 8,10)( 7,11,15)( 9,19,12)(18,20,21), 
    ( 4,21)( 6,16)( 7,13)( 8, 9)(10,18)(11,15)(12,20)(19,22), 
    ( 4,11)( 6,10)( 7, 9)( 8,13)(12,22)(15,21)(16,18)(19,20), 
    ( 4,22)( 6, 7)( 8,18)( 9,10)(11,12)(13,16)(15,20)(19,21), 
    ( 2,14, 5, 3)( 4, 7, 9,10)( 6,22,12,11)( 8,16,20,21)(15,18)(23,24) ] )
  gap> Size(m22a);
  887040
  gap> m22a:=Operation(m22a,[1..22]);m22a.name:="m22a";;
  Group( ( 1, 2)( 5,14)( 6,18)( 7,12)( 8,19)( 9,20)(11,21)(13,22), ( 1, 3)
  ( 4,13, 7,21)( 5,14)( 6, 9,22,11)( 8,16,15,19)(10,18,12,20), ( 4,13)( 6,19)
  ( 7,21)( 8,11)( 9,15)(10,20)(12,18)(16,22), ( 3, 4,14,22)( 5,16,17,13)( 6, 7)
  ( 8,15,11,10)( 9,12,20,18)(19,21), ( 3,17)( 4,16)( 5,14)( 6, 7)( 8,11)(12,18)
  (13,22)(19,21), ( 4,22,16)( 5,17,14)( 6, 8,10)( 7,11,15)( 9,19,12)(18,20,21), 
  ( 4,21)( 6,16)( 7,13)( 8, 9)(10,18)(11,15)(12,20)(19,22), ( 4,11)( 6,10)
  ( 7, 9)( 8,13)(12,22)(15,21)(16,18)(19,20), ( 4,22)( 6, 7)( 8,18)( 9,10)
  (11,12)(13,16)(15,20)(19,21), ( 2,14, 5, 3)( 4, 7, 9,10)( 6,22,12,11)
  ( 8,16,20,21)(15,18) )
  gap> Size(m22a);
  887040
  gap> s:=SylowSubgroup(m22a,2);;
  gap> a:=AgGroup(s);
  Group( g1, g2, g3, g4, g5, g6, g7, g8 )
  gap> n:=Filtered(NormalSubgroups(a),i->Size(i)=16
  >         and IsElementaryAbelian(i));
  [ Subgroup( Group( g1, g2, g3, g4, g5, g6, g7, g8 ), [ g4, g5*g6, g7, g8 ] ), 
    Subgroup( Group( g1, g2, g3, g4, g5, g6, g7, g8 ), 
      [ g1*g7, g2*g3*g7, g5*g6, g8 ] ), Subgroup( Group( g1, g2, g3, g4, g5, 
      g6, g7, g8 ), [ g2*g3, g5*g6, g7, g8 ] ), 
    Subgroup( Group( g1, g2, g3, g4, g5, g6, g7, g8 ), [ g4*g7, g5, g6, g8 ] ), 
    Subgroup( Group( g1, g2, g3, g4, g5, g6, g7, g8 ), 
      [ g4*g7, g5*g7, g6*g7, g8 ] ) ]
  gap> n:=List(n,i->Image(a.bijection,i));;
  gap> e:=Filtered(n,i->IsRegular(i,PermGroupOps.MovedPoints(i)));;
  gap> Length(e);
  1
  gap> e:=e[1];;
  gap> h:=Normalizer(m22a,e);;
  gap> mop:=Operation(m22a,RightCosets(m22a,h),OnRight);;
  gap> DegreeOperation(mop,[1..100]);
  77
  gap> ophom:=OperationHomomorphism(m22a,mop);;
  gap> dp:=DirectProduct(m22a,mop);;
  gap> emb1:=Embedding(m22a,dp,1);;
  gap> emb2:=Embedding(mop,dp,2);;
  gap> diag:=List(m22a.generators,
  >            i->Image(emb1,i)*Image(emb2,Image(ophom,i)));;
  gap> diag:=Group(diag,());;
  gap> diag.name:="M22.2-99";
  "M22.2-99"
  gap> RequirePackage("grape");
  
  Loading  GRAPE 2.31  (GRaph Algorithms using PErmutation groups),
  by L.H.Soicher@qmw.ac.uk.
  
  gap> gamma:=NullGraph(diag,100);
  rec(
    isGraph := true,
    order := 100,
    group := M22.2-99,
    schreierVector := [ -1, 1, 2, 4, 1, 7, 10, 8, 8, 4, 8, 8, 2, 10, 2, 5, 5, 
        8, 7, 10, 7, 4, -2, 10, 6, 6, 1, 10, 8, 2, 1, 4, 4, 10, 10, 1, 4, 2, 4, 
        5, 4, 4, 1, 8, 2, 7, 4, 4, 8, 4, 2, 10, 10, 2, 3, 10, 6, 9, 10, 10, 4, 
        4, 2, 4, 5, 5, 2, 10, 4, 6, 3, 9, 2, 5, 2, 5, 2, 5, 1, 1, 6, 4, 7, 9, 
        8, 4, 3, 3, 6, 3, 5, 10, 5, 2, 2, 8, 10, 2, 4, -3 ],
    adjacencies := [ [  ], [  ], [  ] ],
    representatives := [ 1, 23, 100 ],
    isSimple := true )
  gap> AddEdgeOrbit(gamma,[1,100]);AddEdgeOrbit(gamma,[100,1]);
  gap> hexad:=First(Orbits(h,[1..22]),i->Length(i)=6);
  [ 2, 8, 17, 15, 22, 6 ]
  gap> for i in hexad do AddEdgeOrbit(gamma,[i,23]);
  > AddEdgeOrbit(gamma,[23,i]); od;
  gap> Adjacency(gamma,23);
  [ 2, 6, 8, 15, 17, 22 ]
  gap> stab:=Stabilizer(diag,23);;
  gap> orbs:=Orbits(stab,[24..99]);;
  gap> orbreps:=List(orbs,i->i[1]);
  [ 24, 39 ]
  gap> Intersection(hexad,Adjacency(gamma,24));
  [ 15, 17 ]
  gap> Intersection(hexad,Adjacency(gamma,39));
  [  ]
  gap> AddEdgeOrbit(gamma,[23,39]); AddEdgeOrbit(gamma,[39,23]);
  gap> IsSimpleGraph(gamma);
  true
  gap> Adjacency(gamma,23);
  [ 2, 6, 8, 15, 17, 22, 39, 42, 46, 49, 52, 56, 63, 68, 70, 76, 80, 81, 86, 
    90, 93, 94 ]
  gap> IsDistanceRegular(gamma);
  true
  gap> aug:=AutGroupGraph(gamma);;
  gap> Size(aug);
  88704000
  gap> DisplayCompositionSeries(aug);
  <G> (3 gens, size 88704000)
   | Z(2)
  <S> (2 gens, size 44352000)
   | HS
  <1> (0 gens)
  gap> hs:=DerivedSubgroup(aug);;
  gap> ct:=CharTable("U3(5)");;ct2:=CharTable("U3(5).2");;
  gap> cths:=CharTable("hs");;
  gap> ct.orders;ct2.orders;cths.orders;
  [ 1, 2, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 10 ]
  [ 1, 2, 3, 4, 5, 5, 5, 6, 7, 8, 10, 2, 4, 6, 8, 10, 12, 20, 20 ]
  [ 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8, 8, 10, 10, 11, 11, 12, 15, 20, 
    20 ]
  gap> repeat e1:=Random(hs);until OrderPerm(e1)=12;
  gap> e2:=e1^6;;
  gap> ct2.classes[2];
  525
  gap> cths.orders[21];
  12
  gap> cths.powermap[3][cths.powermap[2][21]];
  2
  gap> cths.classes[2];
  5775
  gap> cnt:=0;;repeat u:=Subgroup(aug,[e1,e2^Random(hs)]);cnt:=cnt+1;
  > until Index(hs,u)=176;cnt;
  26
  gap> hsop:=Operation(hs,CanonicalRightTransversal(hs,u),
  > OnCanonicalCosetElements(hs,u));;
  gap> IsPrimitive(hsop,[1..176]);
  true
  gap> ophom:=OperationHomomorphism(hs,hsop);;
  gap> dp:=DirectProduct(hs,hsop);;
  gap> emb1:=Embedding(hs,dp,1);;
  gap> emb2:=Embedding(hsop,dp,2);;
  gap> diag:=List(hs.generators,i->Image(emb1,i)*Image(emb2,Image(ophom,i)));;
  gap> diag:=Group(diag,());;diag.name:="hs-276";;
  gap> adj:=Adjacency(gamma,1);
  [ 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 
    98, 99, 100 ]
  gap> t:=Stabilizer(diag,[1,adj[1]],OnSets);;
  gap> cnt:=0;;repeat s:=Subgroup(diag,[Random(t),Random(t)]);cnt:=cnt+1;
  > until Size(s)=Size(t);cnt;t:=s;;
  1
  gap> aus:=t.operations.AutomorphismGroup(t);;
  gap> Size(aus);
  241920
  gap> inner:=Subgroup(aus,aus.innerAutomorphisms);;
  gap> Index(aus,inner);
  6
  gap> rt:=RightTransversal(aus,inner);;
  gap> automs:=Filtered(rt,i->i^2 in inner and not i in inner);;
  gap> Length(automs);
  3
  gap> List(automs,i->Order(aus,i));
  [ 2, 2, 2 ]
  gap> ot:=Orbits(t,[1..276]);;
  gap> List(ot,Length);
  [ 2, 56, 42, 56, 120 ]
  gap> PermutationByAutomorphism := function(grp,aut,dom)
  > local op,oh,p,s,sim,simp,rt,rtim,extelm,l1,l2;
  >   # We compute the action on the given domain and transfer the
  >   # automorphism to this permutation action
  >   op:=Operation(grp,dom);
  >   oh:=OperationHomomorphism(grp,op);
  >   aut:=GroupHomomorphismByImages(op,op,op.generators,List(op.generators,
  >          i->Image(oh,Image(aut,PreImagesRepresentative(oh,i)))));
  >   aut.isMapping:=true; # just to save time (otherwise GAP will test that
  >                  # it is indeed a homomorphism, but we know this already)
  > 
  >   # compute stabilizer and images
  >   s:=Stabilizer(op,1);
  >   sim:=Image(aut,s);
  > 
  >   # is the image a stabilizer? It is if it has an orbit of length 1
  >   simp:=Filtered(Orbits(sim,[1..Length(dom)]),i->Length(i)=1);
  >   if Length(simp)=0 then
  >     return false; # it cannot be induced by a permutation action
  >   fi;
  > 
  >   # the permutation can be obtained by the induced action on the right
  >   # cosets.
  >   simp:=simp[1][1]; #image base point
  >   rt:=RightTransversal(op,s);
  >   rtim:=List(rt,i->Image(aut,i));
  >   l1:=List(rt,i->1^i);l2:=List(rtim,i->simp^i);
  > 
  >   # we got the images, make a permutation from them.
  >   if Length(Orbits(grp,dom))=1 then
  >     extelm:=MappingPermListList(l1,l2);
  >   else
  >     # if we have two orbits, we have to ensure they get swapped
  >     extelm:=MappingPermListList(Concatenation(l1,l2),Concatenation(l2,l1));
  >   fi;
  > 
  >   # test whether the computed element indeed fulfills the specifictions
  >   # (This is a safety test only)
  >   if ForAny(op.generators,i->i^extelm<>i^aut) then
  >     Error("something went wrong");
  >   fi;
  > 
  >   # finally move the points acted on to the original domain.
  >   return extelm^MappingPermListList([1..Length(dom)],dom);
  > end;
  function ( grp, aut, dom ) ... end
  gap> 
  gap> lo:=First(ot,i->Length(i)=120);;
  gap> automs:=Filtered(automs,i->PermutationByAutomorphism(t,i,lo)<>false);;
  gap> Length(automs);
  1
  gap> autom:=automs[1];;
  gap> pos:=Filtered([1..Length(ot)],i->Length(ot[i])=56);
  [ 2, 4 ]
  gap> perms:=List(ot{Difference([1..5],pos)},
  >              i->PermutationByAutomorphism(t,autom,i));;
  gap> element:=Product(perms)*PermutationByAutomorphism(t,
  >                            autom,Concatenation(ot{pos}));;
  gap> ot[1];
  [ 1, 79 ]
  gap> 1^element;
  1
  gap> element:=element*(1,79);;
  gap> co3:=Group(Concatenation(diag.generators,[element]),());;
  gap> Size(co3);
  495766656000
  gap> DisplayCompositionSeries(co3);
  <G> (3 gens, size 495766656000)
   | Co(3)
  <1> (0 gens)
  gap> quit;