ENHANCE: Pc automorphisms prepare better for CompatiblePairs

Ensure that many tails of the pcgs of a factor group correspond to
characteristic subgroups. This way `CompatiblePairs` can reduce the orbit
calculation into smaller steps.

lib/grppcaut.gi

@@ -1228,14 +1228,42 @@ local spaceincl,outvecs,l,sub,yet,i,j,k,s,t,new,incl,min,rans,sofar,done,
   return a;
 end;
 
-
+PcgsCharacteristicTails:=function(G,aut)
+local gens,ser,new,pcgs,f,mo,i,j,k,s;
+  gens:=GeneratorsOfGroup(aut);
+  ser:=InvariantElementaryAbelianSeries(G,gens);
+  new:=[G];
+  for i in [2..Length(ser)] do
+    pcgs:=ModuloPcgs(ser[i-1],ser[i]);
+    f:=GF(RelativeOrders(pcgs)[1]);
+    mo:=List(gens,x->List(pcgs,y->ExponentsOfPcElement(pcgs,y^x))*One(f));
+    mo:=GModuleByMats(mo,f);
+    for j in
+      Reversed(Filtered(MTX.BasesCompositionSeries(mo),
+        x->Length(x)<Length(pcgs))) do
+      s:=ser[i];
+      for k in j do
+	s:=ClosureSubgroupNC(s,PcElementByExponents(pcgs,List(k,Int)));
+      od;
+      Add(new,s);
+    od;
+  od;
+  # build pcgs
+  ser:=[];
+  for i in [2..Length(new)] do
+    pcgs:=ModuloPcgs(new[i-1],new[i]);
+    Append(ser,pcgs);
+  od;
+  pcgs:=PcgsByPcSequence(FamilyObj(One(G)),ser);
+  return pcgs;
+end;
 
 InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
     local spec, weights, first, m, pcgsU, F, pcgsF, A, i, s, n, p, H, 
           pcgsH, pcgsN, N, epi, mats, M, autos, ocr, elms, e, list, imgs,
           auto, tmp, hom, gens, P, C, B, D, pcsA, rels, iso, xset,
           gensA, new,as,somechar,scharorb,asAutom,autactbase,
-	  quotimg,eN,field,act,spaces,sporb;
+	  quotimg,eN,field,act,spaces,sporb,npcgs,nM;
 
     asAutom:=function(sub,hom) return Image(hom,sub);end;
 
@@ -1510,7 +1538,21 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
             Info( InfoAutGrp, 2,"compute compatible pairs in group of size ",
                                   Size(A), " x ",Size(B),", ",
 				  Length(GeneratorsOfGroup(D))," generators");
-            C := CompatiblePairs( F, M, D );
+
+            if Size(D)>10^10 then
+	      # translate to different pcgs to make tails A-invariant
+	      npcgs:=PcgsCharacteristicTails(F,A);
+	      C:=GroupWithGenerators(npcgs);
+	      SetPcgs(C,npcgs);
+	      Assert(1,Pcgs(C)=npcgs); # ensure no magic took place
+	      as:=GroupHomomorphismByImagesNC(F,Group(M.generators),
+	            Pcgs(F),M.generators);
+              nM:=rec(field:=M.field,dimension:=M.dimension,
+		      generators:=List(npcgs,x->ImagesRepresentative(as,x)));
+	      C:=CompatiblePairs(C,nM,D);
+	    else
+	      C := CompatiblePairs( F, M, D );
+	    fi;
         else
             #Info( InfoAutGrp, 2,"compute reduced gl ");
             #B := MormalizingReducedGL( spec, s, n, M );

lib/grppcext.gi

@@ -831,6 +831,7 @@ local G, M, Mgrp, oper, A, B, D, translate, gens, genimgs, triso, K, K1,
 	u:=SubgroupNC(G,pcgs{idx});
       until ForAll(GeneratorsOfGroup(u),
 	i->ForAll(GeneratorsOfGroup(A),j->Image(j,i) in u));
+
       Ggens:=InducedPcgsByPcSequence(pcgs,pcgs{idx});
       tup:=M.generators{idx};
 
@@ -866,7 +867,7 @@ local G, M, Mgrp, oper, A, B, D, translate, gens, genimgs, triso, K, K1,
 	# not being perfect is not likely to do a hash conflict
 	pows:=List([0..Length(tup)],x->pows^x);
 
-	tmp:=[D, rec(hashfun:= lst->lst*pows),tup, gens,genimgs, f ];
+	tmp:=[D, rec(hashfun:= lst->lst*pows),tup, gens,newimgs, f ];
 
 	#  use `op' to get in the fake domain with the hashfun
 	tmp := StabilizerOp( D, rec(hashfun:= lst->lst*pows),tup,