From df91fdb89344e71f43d88b68555193b14c059876 Mon Sep 17 00:00:00 2001
From: Alexander Hulpke <hulpke@math.colostate.edu>
Date: Sat, 26 Nov 2022 12:15:54 -0700
Subject: [PATCH] ENHANCE: Improvement to perm rep of fp.

Try to use induced permutation representation, if we cannot find easily
coreless subgroup that finishes coset enumeration.

This resolves #5222
---
 lib/ghomfp.gd |   7 ++++
 lib/ghomfp.gi |   2 +-
 lib/grpfp.gi  | 100 ++++++++++++++++++++++++++++++++++----------------
 3 files changed, 77 insertions(+), 32 deletions(-)

diff --git a/lib/ghomfp.gd b/lib/ghomfp.gd
index 61c33352fc..42f412b24e 100644
--- a/lib/ghomfp.gd
+++ b/lib/ghomfp.gd
@@ -357,3 +357,10 @@ DeclareGlobalFunction("LargerQuotientBySubgroupAbelianization");
 ##  <#/GAPDoc>
 ##
 DeclareGlobalFunction("ProcessEpimorphismToNewFpGroup");
+
+#############################################################################
+##
+#M  InducedRepFpGroup(<hom>,<u> )
+##
+##  induce <hom> def. on <u> up to the full group
+DeclareGlobalFunction("InducedRepFpGroup");
diff --git a/lib/ghomfp.gi b/lib/ghomfp.gi
index d0f9558ce8..d1c08cd35a 100644
--- a/lib/ghomfp.gi
+++ b/lib/ghomfp.gi
@@ -504,7 +504,7 @@ end);
 #M  InducedRepFpGroup(<hom>,<u> )
 ##
 ##  induce <hom> def. on <u> up to the full group
-BindGlobal("InducedRepFpGroup",function(thom,s)
+InstallGlobalFunction(InducedRepFpGroup,function(thom,s)
 local t,w,c,q,chom,tg,hi,u;
   w:=FamilyObj(s)!.wholeGroup;
 
diff --git a/lib/grpfp.gi b/lib/grpfp.gi
index eff13fec0f..e250e5a53a 100644
--- a/lib/grpfp.gi
+++ b/lib/grpfp.gi
@@ -3989,7 +3989,7 @@ InstallGlobalFunction(IsomorphismPermGroupOrFailFpGroup,
 function(arg)
 local mappow, G, max, p, gens, rels, comb, i, l, m, H, t, gen, silent, sz,
   t1, bad, trial, b, bs, r, nl, o, u, rp, eo, rpo, e, e2, sc, j, z,
-  timerFunc;
+  timerFunc,amax,iso,useind;
 
   timerFunc := GET_TIMER_FROM_ReproducibleBehaviour();
 
@@ -4101,36 +4101,70 @@ local mappow, G, max, p, gens, rels, comb, i, l, m, H, t, gen, silent, sz,
     max:=sz*10;
   fi;
 
+  # Do not die on large coset table
+  amax:=max;
+  if max>10^4*sz then
+    max:=10^3*sz;
+  fi;
+
+  useind:=false;
   t1:=timerFunc();
-  bad:=[];
-  i:=1; 
-  while Size(H)<sz and i<=Length(comb) do 
-    trial:=comb[i];
-    if not ForAny(bad,i->IsSubset(i,trial)) then
-      Info(InfoFpGroup,1,"Try subgroup ",trial);
-      t:=CosetTableFromGensAndRels(gens,rels,trial:silent:=true,max:=max );
-      if t<>fail then
-	Info(InfoFpGroup,1,"has index ",IndexCosetTab(t));
-	p:=t{[1,3..Length(t)-1]};
-	Unbind(t);
-	for j in [1..Length(p)] do
-	  p[j]:=PermList(p[j]);
-	od;
-	H:= GroupByGenerators( p );
-	# compute stabilizer chain with size info.
-	if Length(trial)=0 then
-	  # regular is faithful
-	  SetSize(H,sz);
-	else
-	  StabChain(H,rec(limit:=sz));
-	fi;
-      else
-	# note that this subset fails a coset enumeration
-        Add(bad,Set(trial));
+  while max<amax do
+    bad:=[];
+    i:=1; 
+    while Size(H)<sz and i<=Length(comb) do 
+      trial:=comb[i];
+      if not ForAny(bad,i->IsSubset(i,trial)) then
+        Info(InfoFpGroup,1,"Try subgroup ",trial," with ",max);
+        t:=CosetTableFromGensAndRels(gens,rels,trial:silent:=true,max:=max );
+        if t<>fail then
+          Info(InfoFpGroup,1,"has index ",IndexCosetTab(t));
+          p:=t{[1,3..Length(t)-1]};
+          Unbind(t);
+          for j in [1..Length(p)] do
+            p[j]:=PermList(p[j]);
+          od;
+          H:= GroupByGenerators( p );
+          # compute stabilizer chain with size info.
+          if Length(trial)=0 then
+            # regular is faithful
+            SetSize(H,sz);
+          else
+            StabChain(H,rec(limit:=sz));
+          fi;
+
+
+          # try to use induced rep
+          if Size(H)<sz and Size(H)>1 then
+            iso:=IsomorphismFpGroup(SubgroupNC(G,
+              List(trial,x->ElementOfFpGroup(FamilyObj(One(G)),x))
+              ):silent:=true,max:=2*max);
+            H:=Range(iso);
+            t:=IsomorphismPermGroupOrFailFpGroup(H,max);
+            if t<>fail then
+              t:=iso*t;
+              iso:=InducedRepFpGroup(t,Source(iso));
+              H:=Group(List(GeneratorsOfGroup(G),
+                x->ImagesRepresentative(iso,x)));
+              StabChain(H,rec(limit:=sz));
+              if IsAbelian(H) then
+                t:=MinimalFaithfulPermutationRepresentation(H);
+                H:=Group(List(GeneratorsOfGroup(H),
+                  x->ImagesRepresentative(t,x)));
+                StabChain(H,rec(limit:=sz));
+              fi;
+              useind:=true;
+            fi;
+          fi;
+        else
+          # note that this subset fails a coset enumeration
+          Add(bad,Set(trial));
+        fi;
       fi;
-    fi;
 
-    i:=i+1;
+      i:=i+1;
+    od;
+    max:=Minimum(amax,max*10);
   od;
 
   if Size(H)<sz then
@@ -4140,8 +4174,8 @@ local mappow, G, max, p, gens, rels, comb, i, l, m, H, t, gen, silent, sz,
 
   Info(InfoFpGroup,1,"faithful representation of degree ",NrMovedPoints(H));
 
-  # regular case?
-  if Size(H)=NrMovedPoints(H) then
+  # regular case (unless induced)?
+  if Size(H)=NrMovedPoints(H) and not useind then
     t1:=timerFunc()-t1;
     # try to find a cyclic subgroup that gives a faithful rep.
     b:=fail;
@@ -4198,7 +4232,11 @@ local mappow, G, max, p, gens, rels, comb, i, l, m, H, t, gen, silent, sz,
 
   fi;
 
-  p:=SmallerDegreePermutationRepresentation(H:cheap);
+  if NrMovedPoints(H)*10>=Size(H) then
+    p:=SmallerDegreePermutationRepresentation(H);
+  else
+    p:=SmallerDegreePermutationRepresentation(H:cheap);
+  fi;
   # tell the family that we can now compare elements
   SetCanEasilyCompareElements(FamilyObj(One(G)),true);
   SetCanEasilySortElements(FamilyObj(One(G)),true);