#############################################################################
##
##  This file is part of GAP, a system for computational discrete algebra.
##  This file's authors include Alexander Hulpke.
##
##  Copyright of GAP belongs to its developers, whose names are too numerous
##  to list here. Please refer to the COPYRIGHT file for details.
##
##  SPDX-License-Identifier: GPL-2.0-or-later
##
##  This file contains functions that compute the conjugacy classes of a
##  finite group by homomorphic images.
##  Literature: A.H: Conjugacy classes in finite permutation groups via
##  homomorphic images, MathComp
##


# get classes from classical group if possible.
BindGlobal("ClassesFromClassical",function(G)
local H,hom,d,cl;
  if IsPermGroup(G) and (IsNaturalAlternatingGroup(G)  or
    IsNaturalSymmetricGroup(G)) then
    return ConjugacyClasses(G); # there is a method for this
  fi;
  if not IsNonabelianSimpleGroup(PerfectResiduum(G)) then
    return fail;
  fi;
  d:=DataAboutSimpleGroup(PerfectResiduum(G));
  if d.idSimple.series<>"L" then
    return fail;
  fi;
  hom:=EpimorphismFromClassical(G);
  if hom=fail then
    return fail;
  fi;
  # so far matrix classes only implemented for GL/SL
  if not (IsNaturalGL(Source(hom)) or IsNaturalSL(Source(hom))) then
    return fail;
  fi;

  cl:=ClassesProjectiveImage(hom);
  if HasConjugacyClasses(G) then
    cl:=ConjugacyClasses(G); # will have been set
  elif G=Image(hom) then
    cl:=ConjugacyClasses(Image(hom)); # will have been set
  else
    Info(InfoWarning,1,"Weird class storage");
    return fail;
  fi;
  return cl;
end);

#############################################################################
##
#F  ClassRepsPermutedTuples(<g>,<ran>)
##
##  computes representatives of the colourbars with colours selected from
##  <ran>.
BindGlobal("ClassRepsPermutedTuples",function(g,ran)
local anz,erg,pat,pat2,sym,nrcomp,coldist,stab,dc,i,j,k,sum,schieb,lstab,
      stabs,p;
  anz:=NrMovedPoints(g);
  sym:=SymmetricGroup(anz);
  erg:=[];
  stabs:=[];
  for nrcomp in [1..anz] do
    # all sorted colour distributions
    coldist:=Combinations(ran,nrcomp);
    for pat in OrderedPartitions(anz,nrcomp) do
      Info(InfoHomClass,3,"Pattern: ",pat);
      # compute the partition stabilizer
      stab:=[];
      sum:=0;
      for i in pat do
	schieb:=MappingPermListList([1..i],[sum+1..sum+i]);
	sum:=sum+i;
	stab:=Concatenation(stab,
	        List(GeneratorsOfGroup(SymmetricGroup(i)),j->j^schieb));
      od;
      stab:=Subgroup(sym,stab);
      dc:=List(DoubleCosetRepsAndSizes(sym,stab,g),i->i[1]);

      # compute expanded pattern
      pat2:=[];
      for i in [1..nrcomp] do
        for j in [1..pat[i]] do
	  Add(pat2,i);
	od;
      od;

      for j in dc do
	# the new bar's stabilizer
	lstab:=Intersection(g,ConjugateSubgroup(stab,j));
	p:=Position(stabs,lstab);
	if p=fail then
	  Add(stabs,lstab);
	else
	  lstab:=stabs[p];
	fi;
	# the new bar
	j:=Permuted(pat2,j);
        for k in coldist do
	  Add(erg,[List(j,i->k[i]),lstab]);
	od;
      od;
    od;
  od;
  return erg;
end);

#############################################################################
##
#F  ConjugacyClassesSubwreath(<F>,<M>,<n>,<autT>,<T>,<Lloc>,<comp>,<emb>,<proj>)
##
InstallGlobalFunction(ConjugacyClassesSubwreath,
function(F,M,n,autT,T,Lloc,components,embeddings,projections)
local clT,	# classes T
      lcl,	# Length(clT)
      clTR,	# classes under other group (autT,centralizer)
      fus,	# class fusion
      sci,	# |centralizer_i|
      oci,	# |reps_i|
      i,j,k,l,	# loop
      pfus,	# potential fusion
      op,	# operation of F on components
      ophom,	# F -> op
      clF,	# classes of F
      clop,	# classes of op
      bars,	# colour bars
      barsi,    # partial bars
      lallcolors,# |all colors|
      reps,Mproj,centralizers,centindex,emb,pi,varpi,newreps,newcent,
      newcentindex,centimages,centimgindex,C,p,P,selectcen,select,
      cen,eta,newcentlocal,newcentlocalindex,d,dc,s,t,elm,newcen,shift,
      cengen,b1,ore,
      # as in paper
      colourbar,newcolourbar,possiblecolours,potentialbars,bar,colofclass,
      clin,clout,
      etas,	# list of etas
      opfun,	# operation function
      r,rp,	# op-element complement in F
      cnt,
      brp,bcen,
      centralizers_r, # centralizers of r
      newcent_r,# new list to buid
      centrhom, # projection \rest{centralizer of r}
      localcent_r, # image
      cr,
      isdirprod,# is just M a direct product
      genpos,	# generator index
      genpos2,
      gen,	# generator
      stab,	# stabilizer
      stgen,	# local stabilizer generators
      trans,
      repres,
      img,
      limg,
      con,
      pf,
      orb,	# orbit
      orpo,	# orbit position
      minlen,	# minimum orbit length
      remainlen,#list of remaining lengths
      gcd,	# gcd of remaining orbit lengths
      stabtrue,
      diff,
      possible,
      combl,
      smacla,
      smare,
      ppos,
      maxdiff,
      cystr,
      again,	# run orbit again to get all
      trymap,	# operation to try
      skip,	# skip (if u=ug)
      ug,	# u\cap u^{gen^-1}
      scj,	# size(centralizers[j])
      dsz,	# Divisors(scj);
      pper,
      cbs,cbi,
      faillim,
      failcnt;


  Info(InfoHomClass,1,
       "ConjugacyClassesSubwreath called for almost simple group of size ",
        Size(T));
  faillim:=Maximum(100,Size(F)/Size(M));
  isdirprod:=Size(M)=Size(autT)^n;

  # classes of T
  clT:=ClassesFromClassical(T);
  if clT=fail then
    clT:=ConjugacyClassesByRandomSearch(T);
  fi;

  clT:=List(clT,i->[Representative(i),Centralizer(i)]);
  lcl:=Length(clT);
  Info(InfoHomClass,1,"found ",lcl," classes in almost simple");
  clTR:=List(clT,i->ConjugacyClass(autT,i[1]));

  # possible fusion under autT
  fus:=List([1..lcl],i->[i]);
  for i in [1..lcl] do
    sci:=Size(clT[i][2]);
    # we have taken a permutation representation that  prolongates to autT!
    oci:=CycleStructurePerm(clT[i][1]); 

    # we have tested already the smaller-# classes
    pfus:=Filtered([i+1..lcl],j->CycleStructurePerm(clT[j][1])=oci and
      Size(clT[j][2])=sci);
    pfus:=Difference(pfus,fus[i]);
    if Length(pfus)>0 then
      Info(InfoHomClass,3,"possible fusion ",pfus);
      for j in pfus do
        if clT[j][1] in clTR[i] then
	  fus[i]:=Union(fus[i],fus[j]);
	  # fuse the entries
	  for k in fus[i] do
	    fus[k]:=fus[i];
	  od;
	fi;
      od;
    fi;
  od;
  fus:=Set(fus); # throw out duplicates
  colofclass:=List([1..lcl],i->PositionProperty(fus,j->i in j));
  Info(InfoHomClass,2,"fused to ",Length(fus)," colours");

  # get the allowed colour bars
  ophom:=ActionHomomorphism(F,components,OnSets,"surjective");
  op:=Image(ophom);
  lallcolors:=Length(fus);
  bars:=ClassRepsPermutedTuples(op,[1..lallcolors]);

  Info(InfoHomClass,1,"classes in normal subgroup");
  # inner classes
  reps:=[One(M)];
  centralizers:=[M];
  centindex:=[1];
  colourbar:=[[]];

  Mproj:=[];
  varpi:=[];

  for i in [1..n] do
    Info(InfoHomClass,1,"component ",i);
    barsi:=Set(Immutable(List(bars,j->j[1]{[1..i]})));
    emb:=embeddings[i];
    pi:=projections[i];
    Add(varpi,ActionHomomorphism(M,Union(components{[1..i]}),"surjective"));
    Add(Mproj,Image(varpi[i],M));
    newreps:=[];
    newcent:=[];
    newcentindex:=[];
    centimages:=[];
    centimgindex:=[];
    newcolourbar:=[];

    etas:=[]; # etas for the centralizers

    # fuse centralizers that become the same
    for j in [1..Length(centralizers)] do
      C:=Image(pi,centralizers[j]);
      p:=Position(centimages,C);
      if p=fail then
        Add(centimages,C);
	p:=Length(centimages);
      fi;
      Add(centimgindex,p);

      # #force 'centralizers[j]' to have its base appropriate to the component
      # # (this will speed up preimages)
      # cen:=centralizers[j];
      # d:=Size(cen);
      # cen:=Group(GeneratorsOfGroup(cen),());
      # StabChain(cen,rec(base:=components[i],size:=d));
      # centralizers[j]:=cen;
      # etas[j]:=ActionHomomorphism(cen,components[i],"surjective");

    od;
    Info(InfoHomClass,2,Length(centimages)," centralizer images");

    # consider previous centralizers
    for j in [1..Length(centimages)] do
      # determine all reps belonging to this centralizer
      C:=centimages[j];
      selectcen:=Filtered([1..Length(centimgindex)],k->centimgindex[k]=j);
      Info(InfoHomClass,2,"Number ",j,": ",Length(selectcen),
            " previous centralizers to consider");
      
      # 7'
      select:=Filtered([1..Length(centindex)],k->centindex[k] in selectcen);
      # Determine the addable colours
      if i=1 then 
	possiblecolours:=[1..Length(fus)];
      else
	possiblecolours:=[];
	#for k in select do
	#  bar:=colourbar[k];
	k:=1;
	while k<=Length(select) 
	  and Length(possiblecolours)<lallcolors do
	  bar:=colourbar[select[k]];
	  potentialbars:=Filtered(bars,j->j[1]{[1..i-1]}=bar);
	  UniteSet(possiblecolours,
	           potentialbars{[1..Length(potentialbars)]}[1][i]);
	  k:=k+1;
	od;

      fi;

      for k in Union(fus{possiblecolours}) do
	# double cosets
	if Size(C)=Size(T) then
	  dc:=[One(T)];
	else

	  Assert(2,IsSubgroup(T,clT[k][2]));
	  Assert(2,IsSubgroup(T,C));

	  dc:=List(DoubleCosetRepsAndSizes(T,clT[k][2],C),i->i[1]);
	fi;
	for t in selectcen do
	  # continue partial rep. 

#	  #force 'centralizers[j]' to have its base appropriate to the component
#	  # (this will speed up preimages)
#	  if not (HasStabChainMutable(cen) 
#	     and i<=Length(centralizers)
#	     and BaseStabChain(StabChainMutable(cen))[1] in centralizers[i])
#	    then
#	    d:=Size(cen);
#	    cen:= Group( GeneratorsOfGroup( cen ), One( cen ) );
#	    StabChain(cen,rec(base:=components[i],size:=d));
#	    #centralizers[t]:=cen;
#	  fi;

	  cen:=centralizers[t];

	  if not IsBound(etas[t]) then
	    if Number(etas,i->IsBound(i))>500 then
	      for d in
		Filtered([1..Length(etas)],i->IsBound(etas[i])){[1..500]} do
		Unbind(etas[d]);
	      od;
	    fi;
	    etas[t]:=ActionHomomorphism(cen,components[i],"surjective");
	  fi;
	  eta:=etas[t];

	  select:=Filtered([1..Length(centindex)],l->centindex[l]=t);
	  Info(InfoHomClass,3,"centralizer nr.",t,", ",
	       Length(select)," previous classes");
	  newcentlocal:=[];
	  newcentlocalindex:=[];

	  for d in dc do
	    for s in select do
	      # test whether colour may be added here
	      bar:=Concatenation(colourbar[s],[colofclass[k]]);
	      bar:=ShallowCopy(colourbar[s]);
	      Add(bar,colofclass[k]);
	      MakeImmutable(bar);
	      #if ForAny(bars,j->j[1]{[1..i]}=bar) then
	      if bar in barsi then
		# new representative
		elm:=reps[s]*Image(emb,clT[k][1]^d);
		if elm in Mproj[i] then
		  # store the new element
		  Add(newreps,elm);
		  Add(newcolourbar,bar);
		  if i<n then # we only need the centralizer for further 
		              # components
		    newcen:=ClosureGroup(Lloc,
		              List(GeneratorsOfGroup(clT[k][2]),g->g^d));
		    p:=Position(newcentlocal,newcen);
		    if p=fail then
		      Add(newcentlocal,newcen);
		      p:=Length(newcentlocal);
		    fi;
		    Add(newcentlocalindex,p);
		  else
		    Add(newcentlocalindex,1); # dummy, just for counting
		  fi;
		#else
		#  Info(InfoHomClass,5,"not in");
		fi;

	      #else
	      #	Info(InfoHomClass,5,bar,"not minimal");
	      fi;
	      # end the loops from step 9
	    od;
	  od;
	  Info(InfoHomClass,2,Length(newcentlocalindex),
	       " new representatives");

	  if i<n then # we only need the centralizer for further components

	    # Centralizer preimages
	    shift:=[];
	    for l in [1..Length(newcentlocal)] do
	      P:=PreImage(eta,Intersection(Image(eta),newcentlocal[l]));

	      p:=Position(newcent,P);
	      if p=fail then
		Add(newcent,P);
		p:=Length(newcent);
	      fi;
	      shift[l]:=p;
	    od;

	    # move centralizer indices to global
	    for l in newcentlocalindex do
	      Add(newcentindex,shift[l]);
	    od;

	  fi;

	# end the loops from step 6,7 and 8
	od;
      od;
    od;

    centralizers:=newcent;
    centindex:=newcentindex;
    reps:=newreps;
    colourbar:=newcolourbar;
    # end the loop of step 2.
  od;

  Info(InfoHomClass,1,Length(reps)," classreps constructed");

  # allow for faster sorting through color bars
  cbs:=ShallowCopy(colourbar);
  cbi:=[1..Length(cbs)];
  SortParallel(cbs,cbi);

  # further fusion among bars
  newreps:=[];
  Info(InfoHomClass,2,"computing centralizers");
  k:=0;
  for bar in bars do
    k:=k+1;
    #Info(InfoHomClass,4,"k-",k);
    CompletionBar(InfoHomClass,3,"Color Bars ",k/Length(bars));
    b1:=Immutable(bar[1]);
    select:=[];
    i:=PositionSorted(cbs,b1);
    if i<>fail and i<=Length(cbs) and cbs[i]=b1 then
      AddSet(select,cbi[i]);
      while i<Length(cbs) and cbs[i+1]=b1 do
	i:=i+1;
        AddSet(select,cbi[i]);
      od;
    fi;
    #Assert(1,select=Filtered([1..Length(reps)],i->colourbar[i]=b1));

    if Length(select)>1 then
      Info(InfoHomClass,2,"test ",Length(select)," classes for fusion");
    fi;
    newcentlocal:=[];
    for i in [1..Length(select)] do
      if not ForAny(newcentlocal,j->reps[select[i]] in j) then
	#AH we could also compute the centralizer
        C:=Centralizer(F,reps[select[i]]);
	Add(newreps,[reps[select[i]],C]);
	if i<Length(select) and Size(bar[2])>1 then
	  # there are other reps with the same bar left and the bar
	  # stabilizer is bigger than M
	  if not IsBound(bar[2]!.colstabprimg) then
	    # identical stabilizers have the same link. Therefore store the
	    # preimage in them
	    bar[2]!.colstabprimg:=PreImage(ophom,bar[2]);
	  fi;
	  # any fusion would take place in the stabilizer preimage
	  # we know that C must fix the bar, so it is the centralizer there.
	  r:=ConjugacyClass(bar[2]!.colstabprimg,reps[select[i]],C);
	  Add(newcentlocal,r);
	fi;
      fi;
    od;
  od;
  CompletionBar(InfoHomClass,3,"Color Bars ",false);

  Info(InfoHomClass,1,"fused to ",Length(newreps)," inner classes");
  clF:=newreps;
  clin:=ShallowCopy(clF);
  Assert(2,Sum(clin,i->Index(F,i[2]))=Size(M));
  clout:=[];

  # outer classes

  clop:=Filtered(ConjugacyClasses(op),i->Order(Representative(i))>1);

  for k in clop do
    Info(InfoHomClass,1,"lifting class ",Representative(k));

    r:=PreImagesRepresentative(ophom,Representative(k));
    # try to make r of small order
    rp:=r^Order(Representative(k));
    rp:=RepresentativeAction(M,Concatenation(components),
                                  Concatenation(OnTuples(components[1],rp^-1),
				  Concatenation(components{[2..n]})),OnTuples);
    if rp<>fail then
      r:=r*rp;
    else
      Info(InfoHomClass,2,
           "trying random modification to get large centralizer");
      cnt:=LogInt(Size(autT),2)*10;
      brp:=();
      bcen:=Size(Centralizer(F,r));
      repeat
        rp:=Random(M);
	cengen:=Size(Centralizer(M,r*rp));
        if cengen>bcen then
	  bcen:=cengen;
	  brp:=rp;
	  cnt:=LogInt(Size(autT),2)*10;
	else
	  cnt:=cnt-1;
	fi;
      until cnt<0;
      r:=r*brp;
      Info(InfoHomClass,2,"achieved centralizer size ",bcen);
    fi;
    Info(InfoHomClass,2,"representative ",r);
    cr:=Centralizer(M,r);

    # first look at M-action
    reps:=[One(M)];
    centralizers:=[M];
    centralizers_r:=[cr];
    for i in [1..n] do;
      failcnt:=0;
      newreps:=[];
      newcent:=[];
      newcent_r:=[];
      opfun:=function(a,m)
               return Comm(r,m)*a^m;
             end;

      for j in [1..Length(reps)] do
	scj:=Size(centralizers[j]);
	dsz:=0;
	centrhom:=ActionHomomorphism(centralizers_r[j],components[i],
	            "surjective");
	localcent_r:=Image(centrhom);
	Info(InfoHomClass,4,i,":",j);
	Info(InfoHomClass,3,"acting: ",Size(centralizers[j])," minimum ",
	      Int(Size(Image(projections[i]))/Size(centralizers[j])),
	      " orbits.");
	# compute C(r)-classes
	clTR:=[];
	for l in clT do
	  Info(InfoHomClass,4,"DC",Index(T,l[2])," ",Index(T,localcent_r));
	  dc:=DoubleCosetRepsAndSizes(T,l[2],localcent_r);
	  clTR:=Concatenation(clTR,List(dc,i->l[1]^i[1]));
	od;

	orb:=[];
	for p in [1..Length(clTR)] do

	  repres:=PreImagesRepresentative(projections[i],clTR[p]);
	  if i=1 or isdirprod
	     or reps[j]*RestrictedPermNC(repres,components[i]) 
	            in Mproj[i] then
	    stab:=Centralizer(localcent_r,clTR[p]);
	    if Index(localcent_r,stab)<Length(clTR)/10 then
	      img:=Orbit(localcent_r,clTR[p]);
	      #ensure Representative is in first position
	      if img[1]<>clTR[p] then
	        genpos:=Position(img,clTR[p]);
		img:=Permuted(img,(1,genpos));
	      fi;
	    else
	      img:=ConjugacyClass(localcent_r,clTR[p],stab);
	    fi;
	    Add(orb,[repres,PreImage(centrhom,stab),img,localcent_r]);
	  fi;
	od;
	clTR:=orb;

	#was:
	#clTR:=List(clTR,i->ConjugacyClass(localcent_r,i));
	#clTR:=List(clTR,j->[PreImagesRepresentative(projections[i],
	#                                            Representative(j)),
	#	         PreImage(centrhom,Centralizer(j)),
	#		 j]);

	# put small classes to the top (to be sure to hit them and make
	# large local stabilizers)
	SortBy(clTR,x->Size(x[3]));

	Info(InfoHomClass,3,Length(clTR)," local classes");

	cystr:=[];
	for p in [1..Length(clTR)] do
	  repres:=Immutable(CycleStructurePerm(Representative(clTR[p][3])));
	  select:=First(cystr,x->x[1]=repres);
	  if select=fail then
	    Add(cystr,[repres,[p]]);
	  else
	    AddSet(select[2],p);
	  fi;
	od;

	cengen:=GeneratorsOfGroup(centralizers[j]);
	if Length(cengen)>10 then
	  cengen:=SmallGeneratingSet(centralizers[j]);
	fi;
	#cengen:=Filtered(cengen,i->not i in localcent_r);

	while Length(clTR)>0 do

	  # orbit algorithm on classes
	  stab:=clTR[1][2];
	  orb:=[clTR[1]];
	  #repres:=RestrictedPermNC(clTR[1][1],components[i]);
	  repres:=clTR[1][1];
	  trans:=[One(M)];
	  select:=[2..Length(clTR)];

	  orpo:=1;
	  minlen:=Size(orb[1][3]);
	  possible:=false;
	  stabtrue:=false;
	  pf:=infinity;
	  maxdiff:=Size(T);
	  again:=0;
	  trymap:=false;
	  ug:=[];
	  # test whether we have full orbit and full stabilizer
	  while Size(centralizers[j])>(Sum(orb,i->Size(i[3]))*Size(stab)) do
	    genpos:=1;
	    while genpos<=Length(cengen) and
	      Size(centralizers[j])>(Sum(orb,i->Size(i[3]))*Size(stab)) do
	      gen:=cengen[genpos];
	      skip:=false;
	      if trymap<>false then
	        orpo:=trymap[1];
	        gen:=trymap[2];
		trymap:=false;
	      elif again>0 then
		if not IsBound(ug[genpos]) then
		  ug[genpos]:=Intersection(centralizers_r[j],
				   ConjugateSubgroup(centralizers_r[j],gen^-1));
		fi;
		if again<500 and ForAll(GeneratorsOfGroup(centralizers_r[j]),
		          i->i in ug[genpos])
		 then
		  # the random elements will give us nothing new
		  skip:=true;
		else
		  # get an element not in ug[genpos]
		  repeat
		    img:=Random(centralizers_r[j]);
		  until not img in ug[genpos] or again>=500;
		  gen:=img*gen;
	        fi;
	      fi;

	      if not skip then

		img:=Image(projections[i],opfun(orb[orpo][1],gen));

		smacla:=select;

		if not stabtrue then
		  p:=PositionProperty(orb,i->img in i[3]);
		  ppos:=fail;
                else
		  # we have the stabilizer and thus are only interested in
		  # getting new elements.
		  p:=CycleStructurePerm(img);
		  ppos:=First(First(cystr,x->x[1]=p)[2],
			   i->i in select and
			   Size(clTR[i][3])<=maxdiff and img in clTR[i][3]);
		  if ppos=fail then
		    p:="ignore"; #to avoid the first case
		  else
		    p:=fail; # go to first case
		  fi;
		fi;

		if p=fail then
		  if ppos=fail then
		    p:=First(select,
			   i->Size(clTR[i][3])<=maxdiff and img in clTR[i][3]);
		    if p=fail then 
		      return fail;
		    fi;
                  else
		    p:=ppos;
		  fi;

		  RemoveSet(select,p);
		  Add(orb,clTR[p]);

		  if trans[orpo]=false then
		    Add(trans,false);
		  else
		    #change the transversal element to map to the representative
		    con:=trans[orpo]*gen;
		    limg:=opfun(repres,con);
		    con:=con*PreImagesRepresentative(centrhom,
			    RepresentativeAction(localcent_r,
						  Image(projections[i],limg),
						  Representative(clTR[p][3])));
		    Assert(2,Image(projections[i],opfun(repres,con))
			    =Representative(clTR[p][3]));

		    Add(trans,con);

		    for stgen in GeneratorsOfGroup(clTR[p][2]) do
		      Assert( 2, IsOne( Image( projections[i],
				    opfun(repres,con*stgen/con)/repres ) ) );
		      stab:=ClosureGroup(stab,con*stgen/con);
		    od;
		  fi;

		  # compute new minimum length

		  if Length(select)>0 then
		    remainlen:=List(clTR{select},i->Size(i[3]));
		    gcd:=Gcd(remainlen);
		    diff:=minlen-Sum(orb,i->Size(i[3]));

		    if diff<0 then
		      # only go through this if the orbit actually grew
		      # larger
		      minlen:=Sum(orb,i->Size(i[3]));
		      repeat
			if dsz=0 then
			  dsz:=DivisorsInt(scj);
			fi;
			while not minlen in dsz do
                          # workaround rare problem -- try again
                          if First(dsz,i->i>=minlen)=fail then
                            return ConjugacyClassesSubwreath(
                              F,M,n,autT,T,Lloc,components,embeddings,projections);
                          fi;
			  # minimum gcd multiple to get at least the
			  # smallest divisor
			  minlen:=minlen+
			            (QuoInt((First(dsz,i->i>=minlen)-minlen-1),
				            gcd)+1)*gcd;
			od;

			# now try whether we actually can add orbits to make up
			# that length
			diff:=minlen-Sum(orb,i->Size(i[3]));
			Assert(2,diff>=0);
			# filter those remaining classes small enough to make
			# up the length
			smacla:=Filtered(select,i->Size(clTR[i][3])<=diff);
			remainlen:=List(clTR{smacla},i->Size(i[3]));
			combl:=1;
			possible:=false;
			if diff=0 then
			  possible:=fail;
			fi;
			while gcd*combl<=diff 
			      and combl<=Length(remainlen) and possible=false do
			  if NrCombinations(remainlen,combl)<100 then
			    possible:=ForAny(Combinations(remainlen,combl),
					     i->Sum(i)=diff);
			  else
			    possible:=fail;
			  fi;
			  combl:=combl+1;
			od;
			if possible=false then
			  minlen:=minlen+gcd;
			fi;
		      until possible<>false;
		    fi; # if minimal orbit length grew

		    Info(InfoHomClass,5,"Minimum length of this orbit ",
		         minlen," (",diff," missing)");

                  fi;

		  if minlen*Size(stab)=Size(centralizers[j]) then
		    #Assert(1,Length(smacla)>0);
		    maxdiff:=diff;
		    stabtrue:=true;
                  fi;

		elif not stabtrue then
		  # we have an element that stabilizes the conjugacy class.
		  # correct this to an element that fixes the representative.
		  # (As we have taken already the centralizer in
		  # centralizers_r, it is sufficient to correct by
		  # centralizers_r-conjugation.)
		  con:=trans[orpo]*gen;
		  limg:=opfun(repres,con);
		  con:=con*PreImagesRepresentative(centrhom,
			   RepresentativeAction(localcent_r,
						 Image(projections[i],limg),
						 Representative(orb[p][3])));
		  stab:=ClosureGroup(stab,con/trans[p]);
		  if Size(stab)*2*minlen>Size(centralizers[j]) then
		    Info(InfoHomClass,3,
		         "true stabilizer found (cannot grow)");
		    minlen:=Size(centralizers[j])/Size(stab);
		    maxdiff:=minlen-Sum(orb,i->Size(i[3]));
		    stabtrue:=true;
		  fi;
		fi;

		if stabtrue then

		  smacla:=Filtered(select,i->Size(clTR[i][3])<=maxdiff);

		  if Length(smacla)<pf then
		    pf:=Length(smacla);
		    remainlen:=List(clTR{smacla},i->Size(i[3]));

		    CompletionBar(InfoHomClass,3,"trueorb ",1-maxdiff/minlen);
		    #Info(InfoHomClass,3,
		#	"This is the true orbit length (missing ",
		#	maxdiff,")");

		    if Size(stab)*Sum(orb,i->Size(i[3]))
		        =Size(centralizers[j]) then
                      maxdiff:=0;

		    elif Sum(remainlen)=maxdiff then
		      Info(InfoHomClass,2,
			  "Full possible remainder must fuse");
		      orb:=Concatenation(orb,clTR{smacla});
		      select:=Difference(select,smacla);

		    else
		      # test whether there is only one possibility to get
		      # this length
		      if Length(smacla)<20 and
		       Sum(List([1..Minimum(Length(smacla),
		                    Int(maxdiff/gcd+1))],
			   x-> NrCombinations(smacla,x)))<10000 then
			# get all reasonable combinations
			smare:=[1..Length(smacla)]; #range for smacla
			combl:=Concatenation(List([1..Int(maxdiff/gcd+1)],
					      i->Combinations(smare,i)));
			# pick those that have the correct length
			combl:=Filtered(combl,i->Sum(remainlen{i})=maxdiff);
			if Length(combl)>1 then
			  Info(InfoHomClass,3,"Addendum not unique (",
			  Length(combl)," possibilities)");
			  if (maxdiff<10 or again>0) 
			    and ForAll(combl,i->Length(i)<=5) then
			    # we have tried often enough, now try to pick the
			    # right ones 
			    possible:=false;
			    combl:=Union(combl);
			    combl:=smacla{combl};
			    genpos2:=1;
			    smacla:=[];
			    while possible=false and Length(combl)>0 do
			      img:=Image(projections[i],
				opfun(clTR[combl[1]][1],cengen[genpos2]));
			      p:=PositionProperty(orb,i->img in i[3]);
			      if p<>fail then
				# it is!
				Info(InfoHomClass,4,"got one!");

				# remember the element to try
				trymap:=[p,(cengen[genpos2]*
				  PreImagesRepresentative(
				    RestrictedMapping(projections[i],
				      centralizers[j]),
				    RepresentativeAction(
				    orb[p][4],
				    img,Representative(orb[p][3]))  ))^-1];

				Add(smacla,combl[1]);
				combl:=combl{[2..Length(combl)]};
				if Sum(clTR{smacla},i->Size(i[3]))=maxdiff then
				  # bingo!
				  possible:=true;
				fi;
			      fi;
			      genpos2:=genpos2+1;
			      if genpos2>Length(cengen) then
				genpos2:=1;
				combl:=combl{[2..Length(combl)]};
			      fi;
			    od;
			    if possible=false then
			      Info(InfoHomClass,4,"Even test failed!");
			    else
			      orb:=Concatenation(orb,clTR{smacla});
			      select:=Difference(select,smacla);
			      Info(InfoHomClass,3,"Completed orbit (hard)");
			    fi;
			  fi;
			elif Length(combl)>0 then
			  combl:=combl[1];
			  orb:=Concatenation(orb,clTR{smacla{combl}});
			  select:=Difference(select,smacla{combl});
			  Info(InfoHomClass,3,"Completed orbit");
			fi;
		      fi;
		    fi;
		  fi;

	        fi;
	      else
	        Info(InfoHomClass,5,"skip");
	      fi; # if not skip

	      genpos:=genpos+1;
	    od;
	    orpo:=orpo+1;
	    if orpo>Length(orb) then
	      Info(InfoHomClass,3,"Size factor:",EvalF(
	      (Sum(orb,i->Size(i[3]))*Size(stab))/Size(centralizers[j])),
	      " orbit consists of ",Length(orb)," suborbits, iterating");

	      if stabtrue then
		pper:=false;
		# we know stabilizer, just need to find orbit. As these are
		# likely small additions, search in reverse.
		for p in select do
		  for genpos in [1..Length(cengen)] do
		    gen:=Random(centralizers_r[j])*cengen[genpos];
		    img:=Image(projections[i],opfun(clTR[p][1],gen));
		    orpo:=CycleStructurePerm(img);
		    ppos:=First(First(cystr,x->x[1]=orpo)[2],
			   i->not i in select and
			   img in clTR[i][3]);
                    if ppos<>fail and p in select then
		      # so the image is in clTR[ppos] which must be in orb
		      ppos:=Position(orb,clTR[ppos]);
		      Info(InfoHomClass,3,"found new orbit addition ",p);
		      Add(orb,clTR[p]);

#	#change the transversal element to map to the representative
#	con:=trans[ppos]*RepresentativeAction(localcent_r,
#	      Representative(orb[ppos][3]),img)/gen;
#	if not Image(projections[i],opfun(repres,con))
#		=Representative(clTR[p][3]) then
#	  Error("wrong rep");
#	fi;
#	Add(trans,con);
                      # cannot easily do transversal this way.
		      Add(trans,false);

		      RemoveSet(select,p);
		    elif ppos=fail then
		      pper:=true;
		    fi;
		  od;
		od;

		if pper then
		  # trap some weird setup where it does not terminate
		  failcnt:=failcnt+1;
		  if failcnt>=1000*faillim then
		    #Error("fail4");
		    return fail;
		  fi;
		fi;

	      fi;
	     

	      orpo:=1;
	      again:=again+1;
	      if again>1000*faillim then
	        return fail;
	      fi;
	    fi;
	  od;
	  Info(InfoHomClass,2,"Stabsize = ",Size(stab),
		", centstabsize = ",Size(orb[1][2]));
	  clTR:=clTR{select};
	  # fix index positions
	  for p in cystr do
	    p[2]:=Filtered(List(p[2],x->Position(select,x)),IsInt);
	  od;

	  Info(InfoHomClass,2,"orbit consists of ",Length(orb)," suborbits,",
	       Length(clTR)," classes left.");

	  Info(InfoHomClass,3,List(orb,i->Size(i[2])));
	  Info(InfoHomClass,4,List(orb,i->Size(i[3])));

          # select the orbit element with the largest local centralizer
	  orpo:=1;
	  p:=2;
	  while p<=Length(orb) do
	    if IsBound(trans[p]) and Size(orb[p][2])>Size(orb[orpo][2]) then
	      orpo:=p;
	    fi;
	    p:=p+1;
	  od;
	  if orpo<>1 then
	    Info(InfoHomClass,3,"switching to orbit position ",orpo);
	    repres:=opfun(repres,trans[orpo]);
	    #repres:=RestrictedPermNC(clTR[1][1],repres);
	    stab:=stab^trans[orpo];
	  fi;


	  Assert(2,ForAll(GeneratorsOfGroup(stab),
                j -> IsOne( Image(projections[i],opfun(repres,j)/repres) )));

	  # correct stabilizer to element stabilizer
	  Add(newreps,reps[j]*RestrictedPermNC(repres,components[i]));
	  Add(newcent,stab);
	  Add(newcent_r,orb[orpo][2]);
	od;

      od;
      reps:=newreps;
      centralizers:=newcent;
      centralizers_r:=newcent_r;

      Info(InfoHomClass,2,Length(reps)," representatives");
    od;

    select:=Filtered([1..Length(reps)],i->reps[i] in M);
    reps:=reps{select};
    reps:=List(reps,i->r*i);
    centralizers:=centralizers{select};
    centralizers_r:=centralizers_r{select};
    Info(InfoHomClass,1,Length(reps)," in M");

    # fuse reps if necessary
    cen:=PreImage(ophom,Centralizer(k));
    newreps:=[];
    newcentlocal:=[];
    for i in [1..Length(reps)] do
      bar:=CycleStructurePerm(reps[i]);
      ore:=Order(reps[i]);
      newcentlocal:=Filtered(newreps,
		     i->Order(Representative(i))=ore and
		     i!.elmcyc=bar);
      if not ForAny(newcentlocal,j->reps[i] in j) then
        C:=Centralizer(cen,reps[i]);
	# AH can we use centralizers[i] here ? 
	Add(clF,[reps[i],C]);
	Add(clout,[reps[i],C]);
	bar:=ConjugacyClass(cen,reps[i],C);
	bar!.elmcyc:=CycleStructurePerm(reps[i]);
	Add(newreps,bar);
      fi;
    od;
    Info(InfoHomClass,1,"fused to ",Length(newreps)," classes");
  od;
  
  if Sum(clout,i->Index(F,i[2]))<>Size(F)-Size(M) then return fail;fi;

  Info(InfoHomClass,2,Length(clin)," inner classes, total size =",
        Sum(clin,i->Index(F,i[2])));
  Info(InfoHomClass,2,Length(clout)," outer classes, total size =",
        Sum(clout,i->Index(F,i[2])));
  Info(InfoHomClass,3," Minimal ration for outer classes =",
	EvalF(Minimum(List(clout,i->Index(F,i[2])/(Size(F)-Size(M)))),30));

  Info(InfoHomClass,1,"returning ",Length(clF)," classes");

  Assert(2,Sum(clF,i->Index(F,i[2]))=Size(F));
  return clF;

end);

InstallGlobalFunction(ConjugacyClassesFittingFreeGroup,function(G)
local cs,	# chief series of G
      i,	# index cs
      cl,	# list [classrep,centralizer]
      hom,	# G->G/cs[i]
      M,	# cs[i-1]
      N,	# cs[i]
      subN,	# maximan normal in M over N
      csM,	# orbit of nt in M under G
      n,	# Length(csM)
      T,	# List of T_i
      Q,	# Action(G,T)
      Qhom,	# G->Q and F->Q
      S,	# PreImage(Qhom,Stab_Q(1))
      S1,	# Action of S on T[1]
      deg1,	# deg (s1)
      autos,	# automorphism for action
      arhom,	# autom permrep list
      Thom,	# S->S1
      T1,	# T[1] Thom
      w,	# S1\wrQ
      wbas,	# base of w
      emb,	# embeddings of w
      proj,	# projections of wbas
      components, # components of w
      reps,	# List reps in G for 1->i in Q
      F,	# action of G on M/N
      Fhom,	# G -> F
      FQhom,	# Fhom*Qhom
      genimages,# G.generators Fhom
      img,	# gQhom
      gimg,	# gFhom
      act,	# component permcation to 1
      j,k,	# loop
      C,	# Ker(Fhom)
      clF,	# classes of F
      ncl,	# new classes
      FM,	# normal subgroup in F, Fhom(M)
      FMhom,	# M->FM
      dc,	# double cosets
      jim,	# image of j
      Cim,
      CimCl,
      p,
      l,lj,
      l1,
      elm,
      zentr,
      onlysizes,
      good,bad,
      lastM;

  onlysizes:=ValueOption("onlysizes");
  # we assume the group has no solvable normal subgroup. Thus we get all
  # classes by lifts via nonabelian factors and can disregard all abelian
  # factors.

  # we will give classes always by their representatives in G and
  # centralizers by their full preimages in G.

  cs:= ChiefSeriesOfGroup( G );

  # the first step is always simple
  if HasAbelianFactorGroup(G,cs[2]) then
    # try to get the largest abelian factor
    i:=2;
    while i<Length(cs) and HasAbelianFactorGroup(G,cs[i+1]) do
      i:=i+1;
    od;
    cs:=Concatenation([G],cs{[i..Length(cs)]});
    # now cs[1]/cs[2] is the largest abelian factor

    cl:=List(RightTransversal(G,cs[2]),i->[i,G]);
  else
    # compute the classes of the simple nonabelian factor by random search
    hom:=NaturalHomomorphismByNormalSubgroupNC(G,cs[2]);
    cl:=ConjugacyClasses(Image(hom));
    cl:=List(cl,i->[PreImagesRepresentative(hom,Representative(i)),
		    PreImage(hom,StabilizerOfExternalSet(i))]);
  fi;
  lastM:=cs[2];

  for i in [3..Length(cs)] do
    # we assume that cl contains classreps/centralizers for G/cs[i-1]
    # we want to lift to G/cs[i]
    M:=cs[i-1];
    N:=cs[i];
    Info(InfoHomClass,1,i,":",Index(M,N),";  ",Size(N));
    if HasAbelianFactorGroup(M,N) then
      Info(InfoHomClass,2,"abelian factor ignored");
    else
      # nonabelian factor. Now it means real work.

      # 1) compute the action for the factor

      # first, we obtain the simple factors T_i/N.
      # we get these as intersections of the conjugates of the subnormal
      # subgroup

      csM:=CompositionSeries(M); # stored attribute
      if not IsSubset(csM[2],N) then
	# the composition series goes the wrong way. Now take closures of
	# its steps with N to get a composition series for M/N, take the
	# first proper factor for subN.
        n:=3;
	subN:=fail;
	while n<=Length(csM) and subN=fail do
	  subN:=ClosureGroup(N,csM[n]);
	  if Index(M,subN)=1 then
	    subN:=fail; # still wrong
	  fi;
	  n:=n+1;
	od;
      else
	subN:=csM[2];
      fi;
      
      if IsNormal(G,subN) then

	# only one -> Call standard process

	Fhom:=fail;
	# is this an almost top factor?
	if Index(G,M)<10 then
	  Thom:=NaturalHomomorphismByNormalSubgroupNC(G,subN);
	  T1:=Image(Thom,M);
	  S1:=Image(Thom);
	  if Size(Centralizer(S1,T1))=1 then
	    deg1:=NrMovedPoints(S1);
	    Info(InfoHomClass,2,
	      "top factor gives conjugating representation, deg ",deg1);

	    Fhom:=Thom;
	  fi;
	else
	  Thom:=NaturalHomomorphismByNormalSubgroupNC(M,subN);
	  T1:=Image(Thom,M);
	fi;

	if Fhom=fail then
	  autos:=List(GeneratorsOfGroup(G),
		    i->GroupHomomorphismByImagesNC(T1,T1,GeneratorsOfGroup(T1),
		      List(GeneratorsOfGroup(T1),
			    j->Image(Thom,PreImagesRepresentative(Thom,j)^i))));

	  # find (probably another) permutation rep for T1 for which all
	  # automorphisms can be represented by permutations
	  arhom:=AutomorphismRepresentingGroup(T1,autos);
	  S1:=arhom[1];
	  deg1:=NrMovedPoints(S1);
	  Fhom:=GroupHomomorphismByImagesNC(G,S1,GeneratorsOfGroup(G),arhom[3]);
	fi;


	C:=KernelOfMultiplicativeGeneralMapping(Fhom);
	F:=Image(Fhom,G);

	clF:=ClassesFromClassical(F);
	if clF=fail then
	  clF:=ConjugacyClassesByRandomSearch(F);
	fi;

	clF:=List(clF,j->[Representative(j),StabilizerOfExternalSet(j)]);

      else
	csM:=Orbit(G,subN); # all conjugates
	n:=Length(csM);

	if n=1 then
	  Error("this cannot happen");
	  T:=M;
	fi;

	T:=Intersection(csM{[2..Length(csM)]}); # one T_i
	if Length(GeneratorsOfGroup(T))>5 then
	  T:=Group(SmallGeneratingSet(T));
	fi;

	T:=Orbit(G,T); # get all the t's
	# now T[1] is a complement to csM[1] in G/N.
	
	# now compute the operation of G on M/N
	Qhom:=ActionHomomorphism(G,T,"surjective");
	Q:=Image(Qhom,G);
	S:=PreImage(Qhom,Stabilizer(Q,1)); 

	# find a permutation rep. for S-action on T[1]
	Thom:=NaturalHomomorphismByNormalSubgroupNC(T[1],N);
	T1:=Image(Thom);
	if not IsSubset([1..NrMovedPoints(T1)],
	                 MovedPoints(T1)) then
	  Thom:=Thom*ActionHomomorphism(T1,MovedPoints(T1),"surjective");
	fi;
	T1:=Image(Thom,T[1]);
	if IsPermGroup(T1) and
	  NrMovedPoints(T1)>SufficientlySmallDegreeSimpleGroupOrder(Size(T1)) then
	  Thom:=Thom*SmallerDegreePermutationRepresentation(T1:cheap);
	  Info(InfoHomClass,1,"reduced simple degree ",NrMovedPoints(T1),
	    " ",NrMovedPoints(Image(Thom)));
	  T1:=Image(Thom,T[1]);
	fi;

	autos:=List(GeneratorsOfGroup(S),
		  i->GroupHomomorphismByImagesNC(T1,T1,GeneratorsOfGroup(T1),
		    List(GeneratorsOfGroup(T1),
			  j->Image(Thom,PreImagesRepresentative(Thom,j)^i))));

	# find (probably another) permutation rep for T1 for which all
	# automorphisms can be represented by permutations
	arhom:=AutomorphismRepresentingGroup(T1,autos);
	S1:=arhom[1];
	deg1:=NrMovedPoints(S1);
	Thom:=GroupHomomorphismByImagesNC(S,S1,GeneratorsOfGroup(S),arhom[3]);

	T1:=Image(Thom,T[1]);

	# now embed into wreath
	w:=WreathProduct(S1,Q);
	wbas:=DirectProduct(List([1..n],i->S1));
	emb:=List([1..n+1],i->Embedding(w,i));
	proj:=List([1..n],i->Projection(wbas,i));
	components:=WreathProductInfo(w).components;

	# define isomorphisms between the components
	reps:=List([1..n],i->
		PreImagesRepresentative(Qhom,RepresentativeAction(Q,1,i)));

	genimages:=[];
	for j in GeneratorsOfGroup(G) do
	  img:=Image(Qhom,j);
	  gimg:=Image(emb[n+1],img);
	  for k in [1..n] do
	    # look at part of j's action on the k-th factor.
	    # we get this by looking at the action of
	    #   reps[k] *   j    *   reps[k^img]^-1
	    # 1   ->    k  ->  k^img    ->           1
	    # on the first component. 
	    act:=reps[k]*j*(reps[k^img]^-1);
	    # this must be multiplied *before* permuting
	    gimg:=ImageElm(emb[k],ImageElm(Thom,act))*gimg;
	    gimg:=RestrictedPermNC(gimg,MovedPoints(w));
	  od;
	  Add(genimages,gimg);
	od;

	F:=Subgroup(w,genimages);
	if AssertionLevel()>=2 then
	  Fhom:=GroupHomomorphismByImages(G,F,GeneratorsOfGroup(G),genimages);
	  Assert(1,fail<>Fhom);
	else
	  Fhom:=GroupHomomorphismByImagesNC(G,F,GeneratorsOfGroup(G),genimages);
	fi;
	C:=KernelOfMultiplicativeGeneralMapping(Fhom);

	Info(InfoHomClass,1,"constructed Fhom");

	# 2) compute the classes for F

	if n>1 then
          #if IsPermGroup(F) and NrMovedPoints(F)<18 then
	  #  # the old Butler/Theissen approach is still OK
	  #  clF:=[];
	  #  for j in 
	  #   Concatenation(List(RationalClasses(F),DecomposedRationalClass)) do
	  #    Add(clF,[Representative(j),StabilizerOfExternalSet(j)]);
	  #  od;
	  #else
	    FM:=F;
	    for j in components do
	      FM:=Stabilizer(FM,j,OnSets);
	    od;

	    clF:=ConjugacyClassesSubwreath(F,FM,n,S1,
		  Action(FM,components[1]),T1,components,emb,proj);
            if clF=fail then
	      #Error("failer");
	      # weird error happened -- redo
	      j:=Random(SymmetricGroup(MovedPoints(G)));
	      FM:=List(GeneratorsOfGroup(G),x->x^j);
	      F:=Group(FM);
	      SetSize(F,Size(G));
	      FM:=GroupHomomorphismByImagesNC(G,F,GeneratorsOfGroup(G),FM);
	      clF:=ConjugacyClassesFittingFreeGroup(F);
	      clF:=List(clF,x->[PreImagesRepresentative(FM,x[1]),PreImage(FM,x[2])]);
	      return clF;
	    fi;
	  #fi;
	else
	  FM:=Image(Fhom,M);
	  Info(InfoHomClass,1,
	      "classes by random search in almost simple group");

	  clF:=ClassesFromClassical(F);
	  if clF=fail then
	    clF:=ConjugacyClassesByRandomSearch(F);
	  fi;

	  clF:=List(clF,j->[Representative(j),StabilizerOfExternalSet(j)]);
	fi;
      fi; # true orbit of T.

      Assert(2,Sum(clF,i->Index(F,i[2]))=Size(F));
      Assert(2,ForAll(clF,i->Centralizer(F,i[1])=i[2]));

      # 3) combine to form classes of sdp 

      # the length(cl)=1 gets rid of solvable stuff on the top we got ``too
      # early''.
      if IsSubgroup(N,KernelOfMultiplicativeGeneralMapping(Fhom)) then
	Info(InfoHomClass,1,
	    "homomorphism is faithful for relevant factor, take preimages");
	if Size(N)=1 and onlysizes=true then
	  cl:=List(clF,i->[PreImagesRepresentative(Fhom,i[1]),Size(i[2])]);
	else
	  cl:=List(clF,i->[PreImagesRepresentative(Fhom,i[1]),
			    PreImage(Fhom,i[2])]);
        fi;
      else
	Info(InfoHomClass,1,"forming subdirect products");

	FM:=Image(Fhom,lastM);
	FMhom:=RestrictedMapping(Fhom,lastM);
	if Index(F,FM)=1 then
	  Info(InfoHomClass,1,"degenerated to direct product");
	  ncl:=[];
	  for j in cl do
	    for k in clF do
	      # modify the representative with a kernel elm. to project
	      # correctly on the second component
	      elm:=j[1]*PreImagesRepresentative(FMhom,
			  LeftQuotient(Image(Fhom,j[1]),k[1]));
	      zentr:=Intersection(j[2],PreImage(Fhom,k[2]));
	      Assert(3,ForAll(GeneratorsOfGroup(zentr),
		      i->Comm(i,elm) in N));
	      Add(ncl,[elm,zentr]);
	    od;
	  od;

	  cl:=ncl;

	else

	  # first we add the centralizer closures and sort by them
	  # (this allows to reduce the number of double coset calculations)
	  ncl:=[];
	  for j in cl do
	    Cim:=Image(Fhom,j[2]);
	    CimCl:=Cim;
	    #CimCl:=ClosureGroup(FM,Cim); # should be unnecessary, as we took
	    # the full preimage
	    p:=PositionProperty(ncl,i->i[1]=CimCl);
	    if p=fail then
	      Add(ncl,[CimCl,[j]]);
	    else
	      Add(ncl[p][2],j);
	    fi;
	  od;

	  Qhom:=NaturalHomomorphismByNormalSubgroupNC(F,FM);
	  Q:=Image(Qhom);
	  FQhom:=Fhom*Qhom;

	  # now construct the sdp's
	  cl:=[];
	  for j in ncl do
	    lj:=List(j[2],i->Image(FQhom,i[1]));
	    for k in clF do
	      # test whether the classes are potential mates
	      elm:=Image(Qhom,k[1]);
	      if not ForAll(lj,i->RepresentativeAction(Q,i,elm)=fail) then

		#l:=Image(Fhom,j[1]);
		      
		if Index(F,j[1])=1 then
		  dc:=[()];
		else
		  dc:=List(DoubleCosetRepsAndSizes(F,k[2],j[1]),i->i[1]);
		fi;
		good:=0;
		bad:=0;
		for l in j[2] do
		  jim:=Image(FQhom,l[1]);
		  for l1 in dc do
		    elm:=k[1]^l1;
		    if Image(Qhom,elm)=jim then
		      # modify the representative with a kernel elm. to project
		      # correctly on the second component
		      elm:=l[1]*PreImagesRepresentative(FMhom,
				  LeftQuotient(Image(Fhom,l[1]),elm));
		      zentr:=PreImage(Fhom,k[2]^l1);
		      zentr:=Intersection(zentr,l[2]);

		      Assert(3,ForAll(GeneratorsOfGroup(zentr),
			      i->Comm(i,elm) in N));

		      Info(InfoHomClass,4,"new class, order ",Order(elm),
			  ", size=",Index(G,zentr));
		      Add(cl,[elm,zentr]);
		      good:=good+1;
		    else
		      Info(InfoHomClass,5,"not in");
		      bad:=bad+1;
		    fi;
		  od;
		od;
		Info(InfoHomClass,4,good," good, ",bad," bad of ",Length(dc));
	      fi;
	    od;
	  od;
        fi; # real subdirect product

      fi; # else Fhom not faithful on factor

      # uff. That was hard work. We're finally done with this layer.
      lastM:=N;
    fi; # else nonabelian
    Info(InfoHomClass,1,"so far ",Length(cl)," classes computed");
  od;

  if Length(cs)<3 then
    Info(InfoHomClass,1,"Fitting free factor returns ",Length(cl)," classes");
  fi;
  Assert( 2, Sum( List( cl, pair -> Size(G) / Size( pair[2] ) ) ) = Size(G) );
  return cl;
end);

## Lifting code, using new format and compatible with matrix groups

#############################################################################
##
#F  FFClassesVectorSpaceComplement( <N>, <p>, <gens>, <howmuch> )
##
##  This function creates a record  containing information about a complement
##  in <N> to the span of <gens>.
##
# field, dimension, subgenerators (as vectors),howmuch
BindGlobal("FFClassesVectorSpaceComplement",function(field,r, Q,howmuch )
local   zero,  one,  ran,  n,  nan,  cg,  pos,  i,  j,  v;
    
    one:=One( field);  zero:=Zero(field);
    ran:=[ 1 .. r ];
    n:=Length( Q );    nan:=[ 1 .. n ];
    
    cg:=rec( matrix        :=[  ],
	       one           :=one,
               baseComplement:=ShallowCopy( ran ),
               commutator    :=0,
               centralizer   :=0,
               dimensionN    :=r,
               dimensionC    :=n );

    if n = 0  or  r = 0  then
        cg.inverse:=NullMapMatrix;
	cg.projection    :=IdentityMat( r, one );
	cg.needed    :=[];
        return cg;
    fi;
    
    for i  in nan  do
        cg.matrix[ i ]:=Concatenation( Q[ i ], zero * nan );
        cg.matrix[ i ][ r + i ]:=one;
    od;
    TriangulizeMat( cg.matrix );
    pos:=1;
    for v  in cg.matrix  do
        while v[ pos ] = zero  do
            pos:=pos + 1;
        od;
        RemoveSet( cg.baseComplement, pos );
        if pos <= r  then  cg.commutator :=cg.commutator  + 1;
                     else  cg.centralizer:=cg.centralizer + 1;  fi;
    od;

    if howmuch=1 then
      return Immutable(cg);
    fi;

    cg.needed        :=[  ];
    cg.projection    :=IdentityMat( r, one );

    # Find a right pseudo inverse for <Q>.
    Append( Q, cg.projection );
    Q:=MutableTransposedMat( Q );
    TriangulizeMat( Q );
    Q:=TransposedMat( Q );
    i:=1;
    j:=1;
    while i <= r  do
        while j <= n and Q[ j ][ i ] = zero  do
            j:=j + 1;
        od;
        if j <= n and Q[ j ][ i ] <> zero  then
            cg.needed[ i ]:=j;
        else

            # If <Q> does  not  have full rank, terminate when the bottom row
            # is reached.
            i:=r;

        fi;
        i:=i + 1; 
    od;
    
    if IsEmpty( cg.needed )  then
        cg.inverse:=NullMapMatrix;
    else
        cg.inverse:=Q{ n + ran }
                       { [ 1 .. Length( cg.needed ) ] };
	cg.inverse:=ImmutableMatrix(field,cg.inverse,true);
    fi;
    if IsEmpty( cg.baseComplement )  then
        cg.projection:=NullMapMatrix;
    else

        # Find a base change matrix for the projection onto the complement.
        for i  in [ 1 .. cg.commutator ]  do
            cg.projection[ i ][ i ]:=zero;
        od;
        Q:=[  ];
        for i  in [ 1 .. cg.commutator ]  do
            Q[ i ]:=cg.matrix[ i ]{ ran };
        od;
        for i  in [ cg.commutator + 1 .. r ]  do
            Q[ i ]:=ListWithIdenticalEntries( r, zero );
            Q[ i ][ cg.baseComplement[ i-r+Length(cg.baseComplement) ] ]
             :=one;
        od;
        cg.projection:=cg.projection ^ Q;
        cg.projection:=cg.projection{ ran }{ cg.baseComplement };
	cg.projection:=ImmutableMatrix(field,cg.projection,true);
        
    fi;

    return cg;
end);

#############################################################################
##
#F  VSDecompCentAction( <N>, <h>, <C>, <howmuch> )
##
##  Given a homomorphism C -> N, c |-> [h,c],  this function determines (a) a
##  vector space decomposition N =  [h,C] + K with  projection onto K and (b)
##  the  ``kernel'' S <  C which plays   the role of  C_G(h)  in lemma 3.1 of
##  [Mecky, Neub\"user, Bull. Aust. Math. Soc. 40].
##
BindGlobal("VSDecompCentAction",function( pcgs, h, C, field,howmuch )
local   i,  tmp,  v,x,cg;

  i:=One(field);
  x:=List( C, c -> ExponentsOfPcElement(pcgs,Comm( h, c ) )*i);
#Print(Number(x,IsZero)," from ",Length(x),"\n");

  cg:=FFClassesVectorSpaceComplement(field,Length(pcgs),x,howmuch);
  tmp:=[  ];
  for i  in [ cg.commutator + 1 ..
	      cg.commutator + cg.centralizer ]  do
    v:=cg.matrix[ i ];
    tmp[ i - cg.commutator ]:=PcElementByExponentsNC( C,
	      v{ [ cg.dimensionN + 1 ..
		  cg.dimensionN + cg.dimensionC ] } );
  od;
  Unbind(cg.matrix);
  cg.cNh:=tmp;
  return cg;
end);

#############################################################################
##
#F  LiftClassesEANonsolvGeneral( <H>,<N>,<NT>,<cl> )
##
BindGlobal("LiftClassesEANonsolvGeneral",
  function( H, Npcgs, cl, hom, pcisom,solvtriv,fran)
    local  classes,    # classes to be constructed, the result
           correctingelement,
           field,      # field over which <N> is a vector space
	   one,
           h,          # preimage `cl.representative' under <hom>
	   cg,
           cNh,        # centralizer of <h> in <N>
           C,  gens,   # preimage `Centralizer( cl )' under <hom>
           r,          # dimension of <N>
           ran,        # constant range `[ 1 .. r ]'
           aff,        # <N> as affine space
           imgs,  M,   # generating matrices for affine operation
           orb,        # orbit of affine operation
           rep,# set of classes with canonical representatives
           c,  i, # loop variables
	   PPcgs,denomdepths,
	   reduce,
	   corr,
	   correctionfactor,
	   stabfac,
	   stabfacgens,
	   stabfacimg,
	   stabrad,
	   sz,gpsz,subsz,solvsz,
	   orblock,
	   stage,
	   b,j,
	   p,vp,genum,
	   st,gpe,
	   fe,epi,
	   repword,repwords,radidx,img,
	   radrange,farange,
	   ratio,
	   reps,
	   deno,docorrect,
	   k,failcnt,orpo,
	   stabilizergen,stabstack,
	   comm,s,stab;# for class correction

  correctingelement:=function(h,rep,fe)
  local comm;
    comm:=Comm( h, fe ) * Comm( rep, fe );
    comm:= ExponentsOfPcElement(Npcgs,comm)*one;
    ConvertToVectorRep(comm,field);
    comm := List(comm * cg.inverse,Int);
    comm:=PcElementByExponentsNC
      ( Npcgs, Npcgs{ cg.needed }, -comm );
    fe:=fe*comm;
    return fe;
  end;

  h := cl[1];

  field := GF( RelativeOrders( Npcgs )[ 1 ] );
  one:=One(field);
  PPcgs:=ParentPcgs(NumeratorOfModuloPcgs(Npcgs));
  denomdepths:=ShallowCopy(DenominatorOfModuloPcgs(Npcgs)!.depthsInParent);
  Add(denomdepths,Length(PPcgs)+1); # one
  
  # Determine the subspace $[h,N]$ and calculate the centralizer of <h>.
  #cNh := ExtendedPcgs( DenominatorOfModuloPcgs( N!.capH ),
  #               VSDecompCentAction( N, h, N!.capH ) );

  #oldcg:=KernelHcommaC(Npcgs,h,NumeratorOfModuloPcgs(Npcgs),2);

  #cg:=VSDecompCentAction( Npcgs, h, NumeratorOfModuloPcgs(Npcgs),field,2 );
  cg:=VSDecompCentAction( Npcgs, h, Npcgs,field,2 );
#Print("complen =",Length(cg.baseComplement)," of ",cg.dimensionN,"\n");
#if Length(Npcgs)>5 then Error("cb"); fi;

  cNh:=cg.cNh;

  r := Length( cg.baseComplement );
  ran := [ 1 .. r ];
  
  # Construct matrices for the affine operation on $N/[h,N]$.
  Info(InfoHomClass,4,"space=",Size(field),"^",r);
  if Size(field)^r>3*10^8 then Error("too large");fi;
  aff := ExtendedVectors( field ^ r );

  gens:=Concatenation(cl[2],Npcgs,cl[3]); # all generators
  gpsz:=cl[5];

  solvsz:=cl[6];

  radidx:=Length(Npcgs)+Length(cl[2]);

  imgs := [  ];
  for c  in gens  do
    M := [  ];
    for i  in [ 1 .. r ]  do
	M[ i ] := Concatenation( ExponentsConjugateLayer( Npcgs,
	      Npcgs[ cg.baseComplement[ i ] ] , c )
	      * cg.projection, [ Zero( field ) ] );
    od;
    M[ r + 1 ] := Concatenation( ExponentsOfPcElement
			  ( Npcgs, Comm( h, c ) ) * cg.projection,
			  [ One( field ) ] );

    M:=ImmutableMatrix(field,M);
    Add( imgs, M );
  od;

#if Size(field)^r>10^7 then Error("BIG");fi;

  # now compute orbits, being careful to get stabilizers in steps
  #orbreps:=[];
  #stabs:=[];

  orb:=OrbitsRepsAndStabsVectorsMultistage(gens{[1..radidx]},
	imgs{[1..radidx]},pcisom,solvsz,solvtriv,
	gens{[radidx+1..Length(gens)]},
	imgs{[radidx+1..Length(gens)]},cl[4],hom,gpsz,OnRight,aff);

  classes:=[];
  deno:=DenominatorOfModuloPcgs(Npcgs);
  for b in orb do
    rep := PcElementByExponentsNC( Npcgs, Npcgs{ cg.baseComplement },
		    b.rep{ ran } );
    #Assert(3,ForAll(GeneratorsOfGroup(stabsub),i->Comm(i,h*rep) in NT));
    stabrad:=ShallowCopy(b.stabradgens);
#Print("startdep=",List(stabrad,x->DepthOfPcElement(PPcgs,x)),"\n");
#if ForAny(stabrad,x->Order(x)=1) then Error("HUH3"); fi;
    stabfacgens:=b.stabfacgens;
    stabfacimg:=b.stabfacimgs;

    # correct generators. Partially in Pc Image
    if Length(cg.needed)>0 then

      stabrad:=List(stabrad,x->correctingelement(h,rep,x));
      # must make proper pcgs -- correction does not preserve that
      stabrad:=TFMakeInducedPcgsModulo(PPcgs,stabrad,denomdepths);

      # we change by radical elements, so the images in the factor don't
      # change
      stabfacgens:=List(stabfacgens,x->correctingelement(h,rep,x));

    fi;

    correctionfactor:=Characteristic(field)^Length(cg.needed);
    subsz:=b.subsz/correctionfactor;
    c := [h * rep,stabrad,stabfacgens,stabfacimg,subsz,
	   b.stabrsubsz/correctionfactor];
    Assert(3,Size(Group(Concatenation(DenominatorOfModuloPcgs(Npcgs),
       stabrad,stabfacgens)))=subsz);

    Add(classes,c);

  od;

  return classes;

end);

#############################################################################
##
#F  LiftClassesEANonsolvCentral( <H>, <N>, <cl> )
##
# the version for pc groups implicitly uses a pc-group orbit-stabilizer
# algorithm. We can't  do this but have to use a more simple-minded
# orbit/stabilizer approach.
BindGlobal("LiftClassesEANonsolvCentral",
  function( H, Npcgs, cl,hom,pcisom,solvtriv,fran )
local  classes,    	# classes to be constructed, the result
	field,         	# field over which <Npcgs> is a vector space
	o,
	n,r,		# dimensions
	space,
	com,
	comms,
	mats,
	decomp,
	reduce,
	gens,
	radidx,
	stabrad,stabfacgens,stabfacimg,stabrsubsz,relo,orblock,fe,st,
	orb,rep,reps,repword,repwords,p,stabfac,img,vp,genum,gpsz,
	subsz,solvsz,i,j,
	v,
	h,          	# preimage `cl.representative' under <hom>
	C,		# preimage `Centralizer( cl )' under <hom>
	w,    		# coefficient vectors for projection along $[h,N]$
	c;          	# loop variable

  field := GF( RelativeOrders( Npcgs )[ 1 ] );
  h := cl[1];
  #reduce:=ReducedPermdegree(C);
  #if reduce<>fail then
  #  C:=Image(reduce,C);
  #  Info(InfoHomClass,4,"reduced to deg:",NrMovedPoints(C));
  #  h:=Image(reduce,h);
  #  N:=ModuloPcgs(SubgroupNC(C,Image(reduce,NumeratorOfModuloPcgs(N))),
#	          SubgroupNC(C,Image(reduce,DenominatorOfModuloPcgs(N))));
#  fi;

  # centrality still means that conjugation by c is multiplication with
  # [h,c] and that the complement space is generated by commutators [h,c]
  # for a generating set {c|...} of C.

  field:=GF(RelativeOrders(Npcgs)[1]);
  n:=Length(Npcgs);
  o:=One(field);
  stabrad:=Concatenation(cl[2],Npcgs);
  radidx:=Length(stabrad);
  stabfacgens:=cl[3];
  stabfacimg:=cl[4];
  gpsz:=cl[5];
  subsz:=gpsz;
  solvsz:=cl[6];
  stabfac:=TrivialSubgroup(Image(hom));

  gens:=Concatenation(stabrad,stabfacgens); # all generators
  # commutator space basis

  comms:=List(gens,c->o*ExponentsOfPcElement(Npcgs,Comm(h,c)));
  List(comms,x->ConvertToVectorRep(x,field));
  space:=List(comms,ShallowCopy);
  TriangulizeMat(space);
  space:=Filtered(space,i->i<>Zero(i)); # remove spurious columns

  com:=BaseSteinitzVectors(IdentityMat(n,field),space);

  # decomposition of vectors into the subspace basis
  r:=Length(com.subspace);
  if r>0 then
    # if the subspace is trivial, everything stabilizes

    decomp:=Concatenation(com.subspace,com.factorspace)^-1;
    decomp:=decomp{[1..Length(decomp)]}{[1..r]};
    decomp:=ImmutableMatrix(field,decomp);

    # build matrices for the affine action
    mats:=[];
    for w in comms do
      c:=IdentityMat(r+1,o);
      c[r+1]{[1..r]}:=w*decomp; # translation bit
      c:=ImmutableMatrix(field,c);
      Add(mats,c);
    od;

    #subspace affine enumerator
    v:=ExtendedVectors(field^r);

    # orbit-stabilizer algorithm solv/nonsolv version
    #C := Stabilizer( C, v, v[1],GeneratorsOfGroup(C), mats,OnPoints );

    # assume small domain -- so no bother with bitlist
    orb:= [v[1]];
    reps:=[One(gens[1])];
    repwords:=[[]];
    stabrad:=[];
    stabrsubsz:=Size(solvtriv);

    vp:=1;

    for genum in [radidx,radidx-1..1] do
      relo:=RelativeOrders(pcisom!.sourcePcgs)[
	      DepthOfPcElement(pcisom!.sourcePcgs,gens[genum])];
      img:=orb[1]*mats[genum];
      repword:=repwords[vp];
      p:=Position(orb,img);
      if p=fail then
	for j in [1..Length(orb)*(relo-1)] do
	  img:=orb[j]*mats[genum];
	  Add(orb,img);
	  Add(reps,reps[j]*gens[genum]);
	  Add(repwords,repword);
	od;
      else
	rep:=gens[genum]/reps[p];
	Add(stabrad,rep);
	stabrsubsz:=stabrsubsz*relo;
      fi;

    od;
    stabrad:=Reversed(stabrad);

    Assert(1,solvsz=stabrsubsz*Length(orb));

    #nosolvable part
    orblock:=Length(orb);
    vp:=1;
    stabfacgens:=[];
    stabfacimg:=[];
    while vp<=Length(orb) do
      for genum in [radidx+1..Length(gens)] do
	img:=orb[vp]*mats[genum];
	rep:=reps[vp]*gens[genum];
	repword:=Concatenation(repwords[vp],[genum-radidx]);
	p:=Position(orb,img);
	if p=fail then
	  Add(orb,img);
	  Add(reps,rep);
	  Add(repwords,repword);
	  for j in [1..orblock-1] do
	    img:=orb[vp+j]*mats[genum];
    #if img in orb then Error("HUH");fi;
	    Add(orb,img);
	    Add(reps,reps[vp+j]*gens[genum]);
	    # repword stays the same!
	    Add(repwords,repword);
	  od;
	else
	  st:=rep/reps[p];
	  if Length(repword)>0 then
	    # build the factor group element
	    fe:=One(Image(hom));
	    for i in repword do
	      fe:=fe*cl[4][i];
	    od;
	    for i in Reversed(repwords[p]) do
	      fe:=fe/cl[4][i];
	    od;
	    if not fe in stabfac then
	      # not known -- add to generators
	      Add(stabfacgens,st);
	      Add(stabfacimg,fe);
	      stabfac:=ClosureGroup(stabfac,fe);
	    fi;
	  fi;
	fi;
      od;
      vp:=vp+orblock;
    od;

    subsz:=stabrsubsz*Size(stabfac);
  else
    stabrsubsz:=solvsz;
  fi;

  if Length(com.factorspace)=0 then
    classes :=[[h,stabrad,stabfacgens,stabfacimg,subsz,stabrsubsz]];
  else
    classes:=[];
    # enumerator of complement
    v:=field^Length(com.factorspace);
    for w in v do
      c := [h * PcElementByExponentsNC( Npcgs,w*com.factorspace),
	    stabrad,stabfacgens,stabfacimg,subsz,stabrsubsz];
      #if reduce<>fail then
  #	Add(classes,[PreImagesRepresentative(reduce,c[1]),
  #	  PreImage(reduce,c[2])]);
  #      else

  Assert(3,c[6]
    =Size(Group(Concatenation(c[2],DenominatorOfModuloPcgs(Npcgs)))));

      Add(classes,c);
  #      fi;
    od;
  fi;

#  Assert(1,ForAll(classes,i->i[1] in H and IsSubset(H,i[2])));
  return classes;
end);


#############################################################################
##
#F  LiftClassesEATrivRep
##
BindGlobal("LiftClassesEATrivRep",
  function( H, Npcgs, cl, fants,hom, pcisom,solvtriv)
    local  classes,    # classes to be constructed, the result
           h,field,one,solvsz,radidx,gens,imgs,M,bas,
	   gpsz,c,i,npcgsact,usent,dim,found,nsgens,ntgens,nsimgs,mo,
	   basis, ssidx,cpidx,compl,pcgsimgs,pcgssel,
	   sel,pcgs,fasize,nsfgens,fgens,a,norb,fstab,rep,reps,frep,freps,
	   orb,p,rsgens,el,img,j,basinv,newo,orbslev,ssd,result,o,subs,orbsub,
	   sgens,sfgens,z,minvecs,orpo,norpo,maxorb,
	   IteratedMinimizer,OrbitMinimizer,Minimizer,miss;

  npcgsact:=function(c)
    local M,i;
    M := [  ];
    for i  in [ 1 .. dim ]  do
	M[ i ] := ExponentsConjugateLayer( Npcgs,
	Npcgs[ i ] , c )*one;
    od;
    M:=ImmutableMatrix(field,M);
    return M;
  end;

  pcgs:=MappingGeneratorsImages(pcisom)[1];
  field:=GF(RelativeOrders(Npcgs)[1]);
  one:=One(field);
  dim:=Length(Npcgs);

  # action of group
  h := cl[1];
  gens:=Concatenation(cl[2],Npcgs,cl[3]); # all generators
  fgens:=Concatenation(ListWithIdenticalEntries(
	    Length(Npcgs)+Length(cl[2]),One(Range(hom))),cl[4]);
  gpsz:=cl[5];

  solvsz:=cl[6];

  radidx:=Length(Npcgs)+Length(cl[2]);
  imgs := [  ];
  for c  in gens  do
    Add( imgs, npcgsact(c));
  od;
  sel:=Filtered([1..Length(imgs)],x->Order(imgs[x])>1);

  usent:=0;
  found:=0;
  while usent<Length(fants) do
    usent:=usent+1;
    nsfgens:=NormalIntersection(fants[usent],Group(cl[4]));
    fasize:=Size(nsfgens);
    nsfgens:=SmallGeneratingSet(nsfgens);
    nsgens:=List(nsfgens,x->PreImagesRepresentative(hom,x));
    nsimgs:=List(Concatenation(pcgs,nsgens),npcgsact);
    mo:=GModuleByMats(nsimgs,field);
    if not MTX.IsIrreducible(mo) then
      # split space as direct sum under normal sub -- clifford Theory
      o:=MTX.BasesMinimalSubmodules(mo);
      if Length(o)>50 then
        o:=o{Set([1..50],x->Random(1,Length(o)))};
      fi;

      for i in Filtered([1..Length(o)],
	  x->(mo.dimension mod Length(o[x])=0) and Length(o[x])>found) do
	# subspace and images as orbit
	bas:=o[i];
	ssd:=Length(bas);
	if found<ssd and Size(field)^ssd<3*10^7 then
	  Info(InfoHomClass,2,"Trying subspace ",ssd," in ",mo.dimension);
	  orbsub:=Orbit(Group(imgs{sel}),bas,OnSubspacesByCanonicalBasis);
	  if Length(orbsub)*Length(bas)<>Length(bas[1]) then
	    subs:=MTX.InducedActionSubmodule(mo,bas);
	    subs:=MTX.Homomorphisms(subs,mo);
	    orbsub:=Filtered(subs,x->x in orbsub);
	  fi;
	  if Length(orbsub)*Length(bas)=Length(bas[1]) and
  	    RankMat(Concatenation(orbsub))=Length(bas[1]) then
	    found:=ssd;
	    el:=[orbsub,bas,fasize,nsgens,nsimgs,nsfgens,mo];

	    subs:=List([1..Length(orbsub)],x->[(x-1)*ssd+1..x*ssd]);
	    bas:=ImmutableMatrix(field,Concatenation(orbsub)); # this is the new basis
	    basinv:=bas^-1;
  	    Assert(1,basinv<>fail);
	  else
	    Info(InfoHomClass,3,"failed ",Length(orbsub));
	  fi;
	fi;
      od;
    fi;

  od;

  if found=0 then
    Info(InfoHomClass,3,"basic case");
    #Error("BASIC");
    return fail;
  else
    ssd:=found;
    #el is [orbsub,bas,fasize,nsgens,nsimgs,nsfgens,mo];
    orbsub:=el[1];
    bas:=el[2];
    fasize:=el[3];
    nsgens:=el[4];
    nsimgs:=el[5];
    nsfgens:=el[6];
    mo:=el[7];
    Info(InfoHomClass,2,"Using subspace ",ssd," in ",mo.dimension);

    subs:=List([1..Length(orbsub)],x->[(x-1)*ssd+1..x*ssd]);
    bas:=ImmutableMatrix(field,Concatenation(orbsub)); # this is the new basis
    basinv:=bas^-1;
    Assert(1,basinv<>fail);

  fi;

  imgs:=List(imgs,x->bas*x*basinv); # write wrt new basis

  # now determine N-orbits, stepwise

  solvtriv:=Subgroup(Range(pcisom),
      List(DenominatorOfModuloPcgs(Npcgs),x->ImagesRepresentative(pcisom,x)));

  orb:=[rec(len:=1,rep:=Zero(bas[1]),
	stabfacgens:=nsgens,
	stabfacimgs:=nsfgens,
	# only generators in factor
	stabradgens:=Filtered(pcgs,x->not x in DenominatorOfModuloPcgs(Npcgs)),
        stabrsubsz:=Size(Image(pcisom)),
	subsz:=fasize*Product(RelativeOrders(pcgs))
		   )];

  orbslev:=[];
  maxorb:=1;
  for i in [1..Length(subs)] do
    norb:=[];
    el:=Elements(VectorSpace(field,IdentityMat(Length(bas),field){subs[i]}));
    for o in orb do
      newo:= OrbitsRepsAndStabsVectorsMultistage(
	     o.stabradgens,List(o.stabradgens,x->bas*npcgsact(x)*basinv),
	     pcisom,o.stabrsubsz,solvtriv,
	     o.stabfacgens,List(o.stabfacgens,x->bas*npcgsact(x)*basinv),
	     o.stabfacimgs,hom,o.subsz,OnRight,
	     el);
      for j in newo do
	if j.len>maxorb then maxorb:=j.len;fi;
	if i>1 then
	  j.len:=j.len*o.len;
	  j.rep:=Concatenation(o.rep{[1..(i-1)*ssd]},j.rep{[(i-1)*ssd+1..Length(j.rep)]});
	  MakeImmutable(j.rep);
	fi;
	Add(norb,j);
      od;
    od;
    Info(InfoHomClass,3,"Level ",i," , ",Length(norb)," orbits");
    orb:=norb;
    Add(orbslev,ShallowCopy(orb));
  od;

  IteratedMinimizer:=function(vec,allcands)
  local i,stops,a,cands,mapper,fmapper,stabfacgens,stabradgens,stabfacimgs,
	range,lcands,lvec;
    cands:=allcands;
    mapper:=One(Source(hom));
    fmapper:=One(Range(hom));
    stabfacgens:=nsgens;
    stabfacimgs:=nsfgens;
    stabradgens:=pcgs;
    for i in [1..Length(subs)] do
      range:=[1..i*ssd];
      lcands:=Filtered(orbslev[i],
	x->ForAny(cands,y->y.rep{range}=x.rep{range}));
      lvec:=Concatenation(vec{range},Zero(vec{[i*ssd+1..Length(vec)]}));
      result:=OrbitMinimumMultistage(stabradgens,
	   List(stabradgens,x->bas*npcgsact(x)*basinv),
	   pcisom,0,0, # solvable size not needed
	   stabfacgens,
	   List(stabfacgens,x->bas*npcgsact(x)*basinv),
	   stabfacimgs,
	   hom,0, #gpsz not needed
	   OnRight,lvec,maxorb,#Maximum(List(lcands,x->x.len)),
	   Set(List(lcands,x->x.rep)));
      a:=First(lcands,x->x.rep{range}=result.min{range});
      mapper:=mapper*result.elm;
      fmapper:=fmapper*result.felm;
      vec:=vec*bas*npcgsact(result.elm)*basinv; # map vector to so far canonical
      # not all classes are feasible
      Assert(1,ForAny(cands,x->x.rep{range}=vec{range}));
      cands:=Filtered(cands,x->x.rep{range}=vec{range}); 
      stabradgens:=a.stabradgens;
      stabfacgens:=a.stabfacgens;
      stabfacimgs:=a.stabfacimgs;
    od;
    if Length(cands)<>1 then Error("nonunique");fi;
    return rec(elm:=mapper,felm:=fmapper,min:=vec,nclass:=cands[1]);
  end;

  pcgsimgs:=List(pcgs,x->bas*npcgsact(x)*basinv);
  pcgssel:=Filtered([1..Length(pcgs)],x->not IsOne(pcgsimgs[x]));
  nsimgs:=List(nsgens,x->bas*npcgsact(x)*basinv);


  OrbitMinimizer:=function(vec,allcands)
  local a;
    
  if false and allcands[1].len>1 then
    Error();
  fi;
    a:=OrbitMinimumMultistage(pcgs,pcgsimgs,pcisom,0,0,
	nsgens,nsimgs,nsfgens,hom,0,
	OnRight,vec,allcands[1].len,minvecs);
    a.nclass:=First(allcands,x->x.rep=a.min);
    return a;

  end;

  orpo:=NewDictionary(orb[Length(orb)].rep,true,field^Length(orb[1].rep));
  for p in [1..Length(orb)] do
    AddDictionary(orpo,orb[p].rep,p);
  od;

  # now do an orbit algorithm on orb. As the orbit is short no need for
  # two-step. 

  newo:=[];
  while Length(orb)>0 do
    # pick new one
    p:=First([1..Length(orb)],x->IsBound(orb[x]));
    norb:=[orb[p]];
    norpo:=[];
    norpo[p]:=1;
    el:=Filtered(orb,x->x.len=orb[p].len);
    minvecs:=Set(List(el,x->x.rep));
#el:=orbslev[3];
    if orb[p].len>30000 then
      Minimizer:=IteratedMinimizer;
    else
      Minimizer:=OrbitMinimizer;
    fi;

    # as Rad <=N we can assume that the radical part of the stabilizer
    # is known
    rsgens:=ShallowCopy(orb[p].stabradgens);
    a:=Difference([1..Length(gens)],sel);
    sgens:=Concatenation(orb[p].stabfacgens,gens{a});
    sfgens:=Concatenation(orb[p].stabfacimgs,fgens{a});
    fstab:=Group(sfgens);
    reps:=[One(Source(hom))];
    freps:=[One(Range(hom))];
    Unbind(orb[p]);

    # factor missing from stop
    miss:=cl[5]/(norb[1].len*Size(fstab)*norb[1].stabrsubsz);

    i:=1;
    while i<=Length(norb) and miss>1 do
      for j in sel do
	img:=OnRight(norb[i].rep,imgs[j]);
	img:=Minimizer(img,el);

	rep:=reps[i]*gens[j]*img.elm;
	frep:=freps[i]*fgens[j]*img.felm;
	p:=LookupDictionary(orpo,img.min);
	#p:=PositionProperty(norb,x->x.rep=img.min);
	if p=fail then
	  return fail;
	  Error("unknown minimum");
	elif IsBound(norpo[p]) then
	  # old point
	  p:=norpo[p];
	  if miss>=2 then
	Assert(1,norb[i].rep*imgs[j]*bas*npcgsact(img.elm)*basinv=norb[p].rep);
    #Print("A",i," ",j," ",Length(el),"\n");
	    # old point -- stabilize
	    a:=frep/freps[p];
	    if not a in sfgens then
	      Add(sgens,rep/reps[p]);
	      Add(sfgens,a);
	      miss:=miss*Size(fstab);
	      fstab:=ClosureGroup(fstab,a);
	      miss:=miss/Size(fstab);
#Print("miss1:",EvalF(miss)," ",i," of ",Length(norb),"\n");

	    fi;
	  fi;
	else
  #Print("B",i," ",j," ",Length(el),"\n");
	  # new point
	  #p:=PositionProperty(orb,x->x.rep=img.min);
	Assert(1,norb[i].rep*imgs[j]*bas*npcgsact(img.elm)*basinv=orb[p].rep);
	  Add(norb,orb[p]);
	  norpo[p]:=Length(norb);
	  Add(reps,rep);
	  Add(freps,frep);
	  miss:=miss*(Length(norb)-1)/Length(norb);
#Print("miss3:",EvalF(miss)," ",i," of ",Length(norb),"\n");
	  # add conjugate stabilizer
	  #Append(rsgens,List(orb[p].stabradgens,x->rep*x/rep));
	  for z in [1..Length(orb[p].stabfacgens)] do
	    a:=frep*orb[p].stabfacimgs[z]/frep;
	    if not a in fstab then
	      Add(sgens,rep*orb[p].stabfacgens[z]/rep);
	      Add(sfgens,a);
	      miss:=miss*Size(fstab);
	      fstab:=ClosureGroup(fstab,a);
	      miss:=miss/Size(fstab);
#Print("miss2:",miss,"\n");
	    fi;
	  od;
	  Unbind(orb[p]);
	fi;
      od;
      i:=i+1;
    od;
if miss<>1 then 
  # something is dodgy -- fallback to the default algorithm
  return fail;Error("HEH?");fi;
    Info(InfoHomClass,3,"Fused ",Length(norb),"*",norb[1].len," ",
      Number(orb,x->IsBound(x))," left");
    Assert(1,ForAll(rsgens,x->norb[1].rep*bas*npcgsact(x)*basinv=norb[1].rep));
    Assert(1,ForAll(sgens,x->norb[1].rep*bas*npcgsact(x)*basinv=norb[1].rep));
#if ForAny(rsgens,x->Order(x)=1) then Error("HUH5"); fi;

    a:=[h*PcElementByExponents(Npcgs,norb[1].rep*bas),rsgens,sgens,sfgens,
	cl[5]/Length(norb)/norb[1].len, norb[1].stabrsubsz];

#rsgens:=List(rsgens,x->ImageElm(pcisom,x));
#if rsgens<>InducedPcgsByGenerators(FamilyPcgs(Range(pcisom)),rsgens) then
#  Error("nonpcgs!");
#fi;

    Add(newo,a);

  od;
  return newo;
end);

InstallGlobalFunction(ConjugacyClassesViaRadical,function (G)
local r,	#radical
      f,	# G/r
      hom,	# G->f
      pcgs,mpcgs, #(modulo) pcgs
      pcisom,
      gens,
      ser,	# series
      radsize,len,ntrihom,
      mran,nran,fran,
      central,
      fants,
      d,
      solvtriv,
      M,N,	# normal subgrops
      ind,	# indices
      i,	#loop
      new,	# new classes
      cl,ncl;	# classes

  # it seems to be cleaner (and avoids deferring abelian factors) if we
  # factor out the radical first. (Note: The radical method for perm groups
  # stores the nat hom.!)
  ser:=FittingFreeLiftSetup(G);
  radsize:=Product(RelativeOrders(ser.pcgs));
  len:=Length(ser.pcgs);

  if radsize=1 then
    cl:=ConjugacyClassesFittingFreeGroup(G:onlysizes:=false);
    ncl:=[];
    for i in cl do
      r:=ConjugacyClass(G,i[1],i[2]);
      Add(ncl,r);
    od;
    return ncl;
  fi;

  pcgs:=ser.pcgs;
  pcisom:=ser.pcisom;
  fants:=[];

  # store centralizers as rep, centralizer generators in radical,
  # centralizer generators with nontrivial
  # radfactor image, corresponding radfactor images
  # the generators in the radical do not list the generators of the
  # current layer after immediate lifting.

  if radsize=Size(G) then
    # solvable case
    hom:=ser.factorhom;
    d:=MappingGeneratorsImages(hom);
    mran:=Filtered([1..Length(d[2])],x->not IsOne(d[2][x]));
    cl:=[[One(G),[],d[1]{mran},d[2]{mran},Size(G),Size(G)]];
  else
    # nonsolvable
    if radsize>1 then
      hom:=ser.factorhom;
      ntrihom:=true;
      f:=Image(hom);
      # if lift setup is inherited, f might not be trivial-fitting
      if Size(RadicalGroup(f))>1 then
        # this is proper recursion
        cl:=ConjugacyClasses(f:onlysizes:=false);
        cl:=List(cl,x->[Representative(x),Centralizer(x)]);
      else
      # we need centralizers
        cl:=ConjugacyClassesFittingFreeGroup(f:onlysizes:=false);
      fi;
      fants:=Filtered(NormalSubgroups(f),x->Size(x)>1 and Size(x)<Size(f));
    else
      if IsPermGroup(G) then
	hom:=SmallerDegreePermutationRepresentation(G:cheap);
	ntrihom:=not IsOne(hom);;
      else
        hom:=IdentityMapping(G);
	ntrihom:=false;
      fi;
      f:=Image(hom);
      cl:=ConjugacyClassesFittingFreeGroup(f);
    fi;

    if ntrihom then
      ncl:=[];
      for i in cl do
	new:=[PreImagesRepresentative(hom,i[1])];
	if not IsInt(i[2]) then
	  Add(new,[]); # no generators in radical yet
	  gens:=SmallGeneratingSet(i[2]);
	  Add(new,
	    List(gens,x->PreImagesRepresentative(hom,x)));
          Add(new,gens);
	  #TODO: PreImage groups?
	  #Add(new,PreImage(hom,i[2]));
	  Add(new,radsize*Size(i[2]));
	  Add(new,radsize);
	fi;
        Add(ncl,new);
      od;
      cl:=ncl;

    fi;
  fi;

  Assert(3,ForAll(cl,x->x[6]=Size(Group(Concatenation(x[2],pcgs)))));

  for d in [2..Length(ser.depths)] do
    #M:=ser[i-1];
    #N:=ser[i];
    mran:=[ser.depths[d-1]..len];
    nran:=[ser.depths[d]..len];
    fran:=[mran,nran];

    mpcgs:=InducedPcgsByPcSequenceNC(pcgs,pcgs{mran}) mod
           InducedPcgsByPcSequenceNC(pcgs,pcgs{nran});

    central:= ForAll(GeneratorsOfGroup(G),
		i->ForAll(mpcgs,
		  j->DepthOfPcElement(pcgs,Comm(i,j))>=ser.depths[d]));
    
    # abelian factor, use affine methods
    Info(InfoHomClass,1,"abelian factor ",d,": ",
      Product(RelativeOrders(ser.pcgs){mran}), "->",
      Product(RelativeOrders(ser.pcgs){nran})," central:",central);

    ncl:=[];
    solvtriv:=Subgroup(Range(pcisom),
	List(DenominatorOfModuloPcgs(mpcgs),x->ImagesRepresentative(pcisom,x)));
    for i in cl do
      #Assert(2,ForAll(GeneratorsOfGroup(i[2]),j->Comm(i[1],j) in M));
      if (central or ForAll(Concatenation(i[2],i[3]),
		i->ForAll(mpcgs,
		  j->DepthOfPcElement(pcgs,Comm(i,j))>=ser.depths[d])) ) then
	Info(InfoHomClass,3,"central step");
	new:=LiftClassesEANonsolvCentral(G,mpcgs,i,hom,pcisom,solvtriv,fran);
      elif Length(fants)>0 and Order(i[1])=1 then
	# special case for trivial representative
	new:=LiftClassesEATrivRep(G,mpcgs,i,fants,hom,pcisom,solvtriv);
	if new=fail then
	  new:=LiftClassesEANonsolvGeneral(G,mpcgs,i,hom,pcisom,solvtriv,fran);
	fi;
      else
	new:=LiftClassesEANonsolvGeneral(G,mpcgs,i,hom,pcisom,solvtriv,fran);
      fi;
      #Assert(3,ForAll(new,x->x[6]
      #  =Size(Group(Concatenation(x[2],DenominatorOfModuloPcgs(mpcgs))))));

#if ForAny(new,x->x[2]<>TFMakeInducedPcgsModulo(pcgs,x[2],nran)) then Error("HUH6");fi;
#Print(List(new,x->List(x[2],y->DepthOfPcElement(pcgs,y))),"\n");

      #Assert(1,ForAll(new, i->ForAll(GeneratorsOfGroup(i[2]),j->Comm(j,i[1]) in N)));
      ncl:=Concatenation(ncl,new);
      Info(InfoHomClass,2,Length(new)," new classes (",Length(ncl)," total)");
    od;
    cl:=ncl;
    Info(InfoHomClass,1,"Now: ",Length(cl)," classes (",Length(ncl)," total)");
  od;

  if Order(cl[1][1])>1 then
    # the identity is not in first position
    Info(InfoHomClass,2,"identity not first, sorting");
    Sort(cl,function(a,b) return Order(a[1])<Order(b[1]);end);
  fi;

  Info(InfoHomClass,1,"forming classes");
  ncl:=[];
  for i in cl do
    if IsInt(i[2]) then
      r:=ConjugacyClass(G,i[1]);
      SetSize(r,Size(G)/i[2]);
    else
      d:=SubgroupByFittingFreeData(G,i[3],i[4],i[2]);
      Assert(2,Size(d)=i[5]);
      Assert(2,Centralizer(G,i[1])=d);
      SetSize(d,i[5]);
      r:=ConjugacyClass(G,i[1],d);
      SetSize(r,Size(G)/i[5]);
    fi;
    Add(ncl,r);
  od;

  # as this test is cheap, do it always
  if Sum(ncl,Size)<>Size(G) then
    Error("wrong classes");
  fi;

  cl:=ncl;

  return cl;
end);

#############################################################################
##
#F  LiftConCandCenNonsolvGeneral( <H>,<N>,<NT>,<cl> )
##
BindGlobal("LiftConCandCenNonsolvGeneral",
  function( H, Npcgs, reps, hom, pcisom,solvtriv,fran)
    local  nreps,      # new reps to be constructed, the result
           correctingelement,
	   minvec,
	   cano,       # element that will be canonical
	   cl,
           field,      # field over which <N> is a vector space
	   one,
           h,          # preimage `cl.representative' under <hom>
	   cg,
           cNh,        # centralizer of <h> in <N>
           C,  gens,   # preimage `Centralizer( cl )' under <hom>
           r,          # dimension of <N>
           ran,        # constant range `[ 1 .. r ]'
           aff,        # <N> as affine space
           imgs,  M,   # generating matrices for affine operation
           orb,        # orbit of affine operation
           rep,# set of classes with canonical representatives
           c,  i, # loop variables
	   PPcgs,denomdepths,
	   reduce,
	   corr,
	   correctionfactor,
	   censize,cenradsize,
	   stabfac,
	   stabfacgens,
	   stabfacimgs,
	   stabrad,
	   sz,gpsz,subsz,solvsz,
	   orblock,
	   stage,
	   b,j,x,
	   minimal,minstab,mappingelm,
	   p,vp,genum,
	   st,gpe,
	   fe,epi,
	   repword,repwords,radidx,img,
	   radrange,farange,
	   ratio,
	   deno,docorrect,
	   k,failcnt,orpo,
	   stabilizergen,stabstack,
	   sel,
	   comm,s,stab;# for class correction

  correctingelement:=function(h,rep,fe)
  local comm;
    comm:=Comm( h, fe ) * Comm( rep, fe );
    comm:= ExponentsOfPcElement(Npcgs,comm)*one;
    ConvertToVectorRep(comm,field);
    comm := List(comm * cg.inverse,Int);
    comm:=PcElementByExponentsNC
      ( Npcgs, Npcgs{ cg.needed }, -comm );
    fe:=fe*comm;
    return fe;
  end;

  mappingelm:=function(orb,pos)
  local mc,mcf,i;
    mc:=One(Source(hom));
    mcf:=One(Range(hom));
    for i in orb.repwords[pos] do
      mc:=mc*orb.gens[i];
      mcf:=mcf*orb.fgens[i];
    od;
    i:=pos mod orb.orblock;
    if i=0 then i:=orb.orblock;fi;
    mc:=orb.reps[i]*mc;
    return [mc,mcf];
  end;

  # all reps given must have the same canonical representative on the level
  # above. So they also all have the same centralizer and we can use this.
  cl:=reps[1];
  h := cl[3];

  field := GF( RelativeOrders( Npcgs )[ 1 ] );
  one:=One(field);
  PPcgs:=ParentPcgs(NumeratorOfModuloPcgs(Npcgs));
  denomdepths:=ShallowCopy(DenominatorOfModuloPcgs(Npcgs)!.depthsInParent);
  Add(denomdepths,Length(PPcgs)+1); # one
  
  # Determine the subspace $[h,N]$ and calculate the centralizer of <h>.
  #cNh := ExtendedPcgs( DenominatorOfModuloPcgs( N!.capH ),
  #               VSDecompCentAction( N, h, N!.capH ) );

  #oldcg:=KernelHcommaC(Npcgs,h,NumeratorOfModuloPcgs(Npcgs),2);

  #cg:=VSDecompCentAction( Npcgs, h, NumeratorOfModuloPcgs(Npcgs),field,2 );
  cg:=VSDecompCentAction( Npcgs, h, Npcgs,field,2 );
#Print("complen =",Length(cg.baseComplement)," of ",cg.dimensionN,"\n");
#if Length(Npcgs)>5 then Error("cb"); fi;

  cNh:=cg.cNh;

  r := Length( cg.baseComplement );
  ran := [ 1 .. r ];
  
  # Construct matrices for the affine operation on $N/[h,N]$.
  Info(InfoHomClass,4,"space=",Size(field),"^",r);
  if Size(field)^r>3*10^8 then Error("too large");fi;
  aff := ExtendedVectors( field ^ r );

  # Format for cl is:
  # 1:Element, 2: Conjugate element, 3: Element that will be canonical
  # in factor, 4:conjugator, 5:cenpcgs,
  # 6:cenfac, 7:cenfacimgs, 8:censize, 9:cenfacsize

  gens:=Concatenation(cl[5],Npcgs,cl[6]); # all generators
  gpsz:=cl[8];

  solvsz:=cl[8]/cl[9];

  radidx:=Length(Npcgs)+Length(cl[5]);

  imgs := [  ];
  for c  in gens  do
    M := [  ];
    for i  in [ 1 .. r ]  do
	M[ i ] := Concatenation( ExponentsConjugateLayer( Npcgs,
	      Npcgs[ cg.baseComplement[ i ] ] , c )
	      * cg.projection, [ Zero( field ) ] );
    od;
    M[ r + 1 ] := Concatenation( ExponentsOfPcElement
			  ( Npcgs, Comm( h, c ) ) * cg.projection,
			  [ One( field ) ] );

    M:=ImmutableMatrix(field,M);
    Add( imgs, M );
  od;

#if Size(field)^r>10^7 then Error("BIG");fi;

  # now compute orbits, being careful to get stabilizers in steps
  #orbreps:=[];
  #stabs:=[];

  # change reps to list of more general format
  nreps:=[];
  for x in reps do
    p:=rec(list:=x);
    p.vector:=ExponentsOfPcElement(Npcgs,LeftQuotient(h,x[2]))*One(field);
    p.exponents:=ShallowCopy(p.vector);
    ConvertToVectorRep(p.vector,field);
    p.vector:=p.vector*cg.projection;
    Add(p.vector,One(field));
    Add(nreps,p);
  od;
  reps:=nreps; 

  nreps:=[];
  sel:=[1..Length(reps)];
  while Length(sel)>0 do
    p:=sel[1]; # the one to do
    RemoveSet(sel,p);
    orb:=OrbitsRepsAndStabsVectorsMultistage(gens{[1..radidx]},
	  imgs{[1..radidx]},
	  pcisom,solvsz,solvtriv,
	  gens{[radidx+1..Length(gens)]},
	  imgs{[radidx+1..Length(gens)]},reps[p].list[7],hom,gpsz,OnRight,
	  aff:orbitseed:=reps[p].vector);
    orb:=orb[1];
    # find minimal element, mapper, stabilizer of minimal element
    minvec:=Minimum(orb.orbit);
    minimal:=mappingelm(orb,Position(orb.orbit,minvec));

    # get the real minimum, including N-Orbit
    corr:=ExponentsOfPcElement(Npcgs,
	                       LeftQuotient(h,reps[p].list[2]^minimal[1]));
    corr:=PcElementByExponentsNC(Npcgs,Npcgs{cg.needed},-corr*cg.inverse);
    minimal[1]:=minimal[1]*corr; # real minimizer

    # element that will be the canonical representative in the factor (tail
    # zeroed out)
    corr:=ExponentsOfPcElement(Npcgs,
	   LeftQuotient(h,reps[p].list[2]^minimal[1]));
    cano:=h*PcElementByExponents(Npcgs,corr);

    # this will not be a pcgs, but we induce later anyhow
    stabrad:=List(orb.stabradgens,x->x^minimal[1]); 
    stabfacgens:=List(orb.stabfacgens,x->x^minimal[1]);
    stabfacimgs:=List(orb.stabfacimgs,x->x^minimal[2]);

    censize:=orb.subsz;
    cenradsize:=orb.stabrsubsz;

    # correct generators. Partially in Pc Image
    if Length(cg.needed)>0 then

      rep:=LeftQuotient(h,reps[p].list[2]^minimal[1]);

      stabrad:=List(stabrad,x->correctingelement(h,rep,x));
      # must make proper pcgs -- correction does not preserve that
      stabrad:=TFMakeInducedPcgsModulo(PPcgs,stabrad,denomdepths);

      # we change by radical elements, so the images in the factor don't
      # change
      stabfacgens:=List(stabfacgens,x->correctingelement(h,rep,x));

      correctionfactor:=Characteristic(field)^Length(cg.needed);
      censize:=censize/correctionfactor;
      cenradsize:=cenradsize/correctionfactor;
    fi;

    # Format for cl is:
    # 1:Element, 2: Conjugate element, 3: Element that will be canonical
    # in factor, 4:conjugator, 5:cenpcgs,
    # 6:cenfac, 7:cenfacimgs, 8:censize, 9:cenfacsize
 
    Add(nreps,[reps[p].list[1],
               reps[p].list[2]^minimal[1],
	       cano,
	       reps[p].list[4]*minimal[1],
	       stabrad,stabfacgens,stabfacimgs,
	       censize,censize/cenradsize]);

    for cl in sel do
      b:=Position(orb.orbit,reps[cl].vector);
      if b<>fail then
	RemoveSet(sel,cl);
	# now find rep mapping 1 here
	b:=mappingelm(orb,b);
	b:=[LeftQuotient(b[1],minimal[1]),
			      LeftQuotient(b[2],minimal[2])];

	# get the real minimum, including N-Orbit
	corr:=ExponentsOfPcElement(Npcgs,
				  LeftQuotient(h,reps[cl].list[2]^b[1]));
	corr:=PcElementByExponentsNC(Npcgs,Npcgs{cg.needed},corr*cg.inverse);
	b[1]:=b[1]/corr; # real minimizer

	# Format for cl is:
	# 1:Element, 2: Conjugate element, 3: Element that will be canonical
	# in factor, 4:conjugator, 5:cenpcgs,
	# 6:cenfac, 7:cenfacimgs, 8:censize, 9:cenfacsiz
	Add(nreps,[reps[cl].list[1],
		  reps[cl].list[2]^b[1],
		  cano,
		  reps[cl].list[4]*b[1],
		  stabrad,stabfacgens,stabfacimgs,
		  censize,censize/cenradsize]);

      elif ValueOption("conjugacytest")=true then
	# in conj test this would mean fail
	return fail;
      fi;
    od;
  od;

  return nreps;

end);

# canonical rep/centralizer
BindGlobal("TFCanonicalClassRepresentative",function (G,candidates)
local r,	#radical
      f,	# G/r
      hom,	# G->f
      prereps,  # fixed factor class reps preimages
      pcgs,mpcgs, #(modulo) pcgs
      pcisom,
      gens,
      ser,	# series
      radsize,len,ntrihom,
      mran,nran,fran,
      central,
      #fants,
      reps,
      nreps,
      fr,
      conj,
      d,
      solvtriv,
      select,sel,pos,
      M,N,	# normal subgrops
      ind,	# indices
      i,j,	#loop
      new,	# new classes
      classrange,
      cl,ncl;	# classes

  # it seems to be cleaner (and avoids deferring abelian factors) if we
  # factor out the radical first. (Note: The radical method for perm groups
  # stores the nat hom.!)
  ser:=FittingFreeLiftSetup(G);
  radsize:=Product(RelativeOrders(ser.pcgs));
  len:=Length(ser.pcgs);

  pcgs:=ser.pcgs;
  pcisom:=ser.pcisom;
  #fants:=fail;

  # store centralizers as rep, centralizer generators in radical,
  # centralizer generators with nontrivial
  # radfactor image, corresponding radfactor images
  # the generators in the radical do not list the generators of the
  # current layer after immediate lifting.

  if radsize=Size(G) then
    # solvable case
    hom:=ser.factorhom;
    d:=MappingGeneratorsImages(hom);
    mran:=Filtered([1..Length(d[2])],x->not IsOne(d[2][x]));
    # elm, elmconj, canonical factor, conjugator, cenpcgs,cenfac,cenfacimgs,censize,cenfacsiz
    reps:=List(candidates,x->[x,x,One(G),One(G),[],d[1]{mran},d[2]{mran},Size(G),Size(G)]);
  else
    # nonsolvable
    if radsize>1 then
      hom:=ser.factorhom;
      ntrihom:=true;
      f:=Image(hom);
      # we need centralizers
      #fants:=Filtered(NormalSubgroups(f),x->Size(x)>1 and Size(x)<Size(f));
    else
      if IsPermGroup(G) then
	hom:=SmallerDegreePermutationRepresentation(G:cheap);
	ntrihom:=not IsOne(hom);;
      else
        hom:=IdentityMapping(G);
	ntrihom:=false;
      fi;
      f:=Image(hom);
    fi;
    if not IsBound(f!.someClassReps) then
      f!.someClassReps:=[ConjugacyClass(f,One(f))]; # identity first
    fi;
    if HasConjugacyClasses(f) and
      Length(f!.someClassReps)<Length(ConjugacyClasses(f)) then
      # expand the list of stored factor classes for once.
      cl:=Filtered(ConjugacyClasses(f),x-> not ForAny(f!.someClassReps,
	   y->Order(Representative(x))=Order(Representative(y))
	   and Representative(y) in x));
      Append(f!.someClassReps,cl);
    fi;
    cl:=f!.someClassReps;

    classrange:=[1..Length(cl)];
    r:=ValueOption("candidatenums");
    if r<>fail and HasConjugacyClasses(G) then
      # candidatenums gives the numbers of some classes in G that should be
      # tried first (as they likely contain the element). Us this to reduce
      # conjugacy test in factor.
      if not IsBound(G!.radicalfactorclassmap) then
	G!.radicalfactorclassmap:=[];
      fi;
      fr:=G!.radicalfactorclassmap;
      for i in Filtered(r,x->not IsBound(fr[x])) do
	# compute images that are not yet known
	d:=ImagesRepresentative(hom,Representative(ConjugacyClasses(G)[i]));
	j:=First([1..Length(cl)],x->IsBound(cl[x]) and d in cl[x]);
	if j<>fail then
	  fr[i]:=j;
	fi;
      od;
      r:=Filtered(r,x->IsBound(fr[x]));
      r:=Set(fr{r}); # class numbers in factor
      classrange:=Concatenation(r,
		   Filtered(classrange,x->not x in r));

    fi;

    nreps:=[];
    for i in candidates do
      fr:=ImagesRepresentative(hom,i);
      conj:=fail;
      j:=0;
      while conj=fail and j<Length(classrange) do
	j:=j+1;
        if Order(fr)=Order(Representative(cl[classrange[j]])) then
	  conj:=RepresentativeAction(f,fr,Representative(cl[classrange[j]]));
	fi;
      od;

      if conj=fail then
	# not yet found, and classes of f were not known -- store this rep
	# image as canonical one for future use.
	j:=j+1;
	Add(cl,ConjugacyClass(f,fr));
        Add(classrange,Length(cl));
	conj:=One(f);
      else
	j:=classrange[j];
      fi;
      # store fixed preimages of reps to avoid any impact of homomorphism.
      if not IsBound(f!.classpreimgs) then
	f!.classpreimgs:=[];
      fi;
      prereps:=f!.classpreimgs;
      if not IsBound(prereps[j]) then
	prereps[j]:=PreImagesRepresentative(hom,Representative(cl[j]));
      fi;

      r:=PreImagesRepresentative(hom,conj);
      d:=GeneratorsOfGroup(Centralizer(cl[j]));
      # Format for cl is:
      # 1:Element, 2: Conjugate element, 3: Element that will be canonical
      # in factor, 4:conjugator, 5:cenpcgs,
      # 6:cenfac, 7:cenfacimgs, 8:censize, 9:cenfacsize
      Add(nreps,[i,i^r,prereps[j],r,[],
        List(d,x->PreImagesRepresentative(hom,x)),d,
	radsize*Size(Centralizer(cl[j])), Size(Centralizer(cl[j]))]);
    od;
    reps:=nreps;

  fi;

  for d in [2..Length(ser.depths)] do
    #M:=ser[i-1];
    #N:=ser[i];
    mran:=[ser.depths[d-1]..len];
    nran:=[ser.depths[d]..len];
    fran:=[mran,nran];

    mpcgs:=InducedPcgsByPcSequenceNC(pcgs,pcgs{mran}) mod
           InducedPcgsByPcSequenceNC(pcgs,pcgs{nran});

    central:= ForAll(GeneratorsOfGroup(G),
		i->ForAll(mpcgs,
		  j->DepthOfPcElement(pcgs,Comm(i,j))>=ser.depths[d]));
    
    # abelian factor, use affine methods
    Info(InfoHomClass,1,"abelian factor ",d,": ",
      Product(RelativeOrders(ser.pcgs){mran}), "->",
      Product(RelativeOrders(ser.pcgs){nran})," central:",central);

    nreps:=[];
    solvtriv:=Subgroup(Range(pcisom),
	List(DenominatorOfModuloPcgs(mpcgs),x->ImagesRepresentative(pcisom,x)));
    select:=[1..Length(reps)];
    while Length(select)>0 do
      pos:=select[1];
      sel:=Filtered(select,x->reps[x][3]=reps[pos][3]);
      if ValueOption("conjugacytest")=true and Length(sel)<>2 then
	return fail;
      fi;
      Info(InfoHomClass,2,Length(sel)," in candidate group");
      select:=Difference(select,sel);
      new:=LiftConCandCenNonsolvGeneral(G,mpcgs,reps{sel},hom,pcisom,
             solvtriv,fran);
      # conj test 
      if new=fail then
	return new;
      fi;
      Append(nreps,new);
    od;
    reps:=nreps;
  od;

  # arrange back to same ordering as before.
  nreps:=[];
  for i in candidates do
    Add(nreps,First(reps,x->IsIdenticalObj(x[1],i)));
  od;

  return nreps;
end);

#############################################################################
##
#M  Centralizer( <G>, <e> ) . . . . . . . . . . . . . . using TF method
##
InstallMethod( CentralizerOp, "TF method:elm",IsCollsElms,
  [ IsGroup and IsFinite and HasFittingFreeLiftSetup,
  IsMultiplicativeElementWithInverse ], OVERRIDENICE,
function( G, e )
  if IsPermGroup(G) or IsPcGroup(G) then TryNextMethod();fi;
  if not e in G then 
    # TODO: form larger group containing e.
    TryNextMethod();
  fi;
  e:=TFCanonicalClassRepresentative(G,[e])[1];
  if e=fail then TryNextMethod();fi;
  return SubgroupByFittingFreeData(G,e[6],e[7],e[5]);
end );

#############################################################################
##
#M  Centralizer( <G>, <e> ) . . . . . . . . . . . . . . using TF method
##
InstallMethod( CentralizerOp, "TF method:subgroup",IsIdenticalObj,
  [ IsGroup and IsFinite and HasFittingFreeLiftSetup,
  IsGroup and IsFinite and HasGeneratorsOfGroup], 
  2*OVERRIDENICE,
function( G, S )
local c,e;
  if IsPermGroup(G) or IsPcGroup(G) then TryNextMethod();fi;
  c:=G;
  for e in GeneratorsOfGroup(S) do
    c:=Centralizer(c,e);
  od;
  return c;
end );

#############################################################################
##
#M  RepresentativeAction( <G>, <d>, <e>, <act> ) . . . . . using TF method
##
InstallOtherMethod( RepresentativeActionOp, "TF Method on elements",
  IsCollsElmsElmsX, 
  [ IsGroup and IsFinite and HasFittingFreeLiftSetup,
        IsMultiplicativeElementWithInverse,
	IsMultiplicativeElementWithInverse, IsFunction ], 
  OVERRIDENICE,
function ( G, d, e, act )
local c;
  if IsPermGroup(G) or IsPcGroup(G) then TryNextMethod();fi;
  if not (d in G and e in G) then 
    # TODO: form larger group containing e.
    TryNextMethod();
  fi;

  if act=OnPoints then #and d in G and e in G then
    c:=TFCanonicalClassRepresentative(G,[d,e]:conjugacytest);
    if c=fail then
      return fail;
    else
      if c[1][2]=c[2][2] then
	return c[1][4]/c[2][4]; # map via canonicals
      else
	return fail; # not conjugate
      fi;
    fi;
  fi;
  TryNextMethod();
end);

#############################################################################
##
#M  ConjugacyClasses( <G> ) . . . . . . . . . . . . . . . . . . of perm group
##
InstallMethod( ConjugacyClasses, "perm group", true, 
  [ IsPermGroup and IsFinite],OVERRIDENICE,
function( G )
local cl;
  if IsNaturalSymmetricGroup(G) or IsNaturalAlternatingGroup(G) then
    # there are better methods for Sn/An
    TryNextMethod();
  fi;

  cl:=ConjugacyClassesForSmallGroup(G);
  if cl<>fail then
    return cl;
  elif IsSolvableGroup( G ) and CanEasilyComputePcgs(G) then
    return ConjugacyClassesForSolvableGroup(G);
  elif IsNonabelianSimpleGroup( G ) then
    cl:=ClassesFromClassical(G);
    if cl=fail then
      cl:=ConjugacyClassesByRandomSearch( G );
    fi;
    return cl;
  else
    return ConjugacyClassesViaRadical(G);
  fi;
end );

#############################################################################
##
#M  ConjugacyClasses( <G> ) . . . . . . . . . . . . . . . . . . of perm group
##
InstallMethod( ConjugacyClasses, "TF Method", true,
  [ IsGroup and IsFinite and CanComputeFittingFree],OVERRIDENICE,
function(G)
  if IsPermGroup(G) or IsPcGroup(G) then TryNextMethod();fi;
  return ConjugacyClassesViaRadical(G);
end);
