ENHANCE: Better strategy for cyclic subgroup for fp group order

Larger cyclic subgroup order will speed up MTC and thus in turn computation
of order of fp group.

lib/grpfp.gi

@@ -3726,7 +3726,7 @@ end );
 ## enumerations with cumulatively bigger coset tables up to table size
 ## <maxtable>. It returns `fail' if no table could be found.
 BindGlobal("FinIndexCyclicSubgroupGenerator",function(G,maxtable)
-local fgens,grels,max,gens,t,Attempt;
+local fgens,grels,max,gens,t,Attempt,perms,short;
  fgens:=FreeGeneratorsOfFpGroup(G);
  grels:=RelatorsOfFpGroup(G);
  max:=ValueOption("max");
@@ -3742,6 +3742,7 @@ local fgens,grels,max,gens,t,Attempt;
  #pseudorandom element - try if it works
  PseudoRandom(G:radius:=Random(2,3))],
  Filtered(gens,j->UnderlyingElement(j)<>UnderlyingElement(t)));
+ gens:=Set(List(gens,UnderlyingElement));
 
  # recursive search (via smaller and smaller partitions) for a finite index
  # subgroup
@@ -3753,7 +3754,7 @@ local fgens,grels,max,gens,t,Attempt;
  trial:=sgens{[1..m]};
  Info(InfoFpGroup,1,"FIS: trying ",trial);
  t:=CosetTableFromGensAndRels(fgens,grels,
- List(trial,UnderlyingElement):silent:=true,max:=max);
+ trial:silent:=true,max:=max);
  if t<>fail and Length(trial)>1 then
  Unbind(t);
  t:=Attempt(trial);
@@ -3780,7 +3781,17 @@ local fgens,grels,max,gens,t,Attempt;
  while max<=maxtable do
  t:=Attempt(gens);
  if t<>fail then
- return t;
+ perms:=List(t[2]{[1,3..Length(t[2])-1]},PermList);
+ short:=FreeGeneratorsOfFpGroup(G);
+ short:=Concatenation(short, List(short,Inverse));
+ short:=Set(List(Concatenation(List([1..3],x->Arrangements(short,x))),
+ Product));
+ short:=List(short,
+ x->[Order(MappedWord(x,FreeGeneratorsOfFpGroup(G),perms)),x]);
+ Sort(short);
+ Info(InfoFpGroup,1,"FIS: better ",short[Length(short)][1]);
+ return [ElementOfFpGroup(FamilyObj(One(G)),short[Length(short)][2]),
+ max];
  fi;
  if max*3/2<maxtable and max*2>maxtable then
  max:=maxtable;
@@ -3829,7 +3840,7 @@ local fgens, # generators of the free group
  # the group could be quite big -- try to find a cyclic subgroup of
  # finite index.
  gen:=FinIndexCyclicSubgroupGenerator(G,infinity);
- max:=gen[3];
+ max:=gen[2];
  gen:=gen[1];
 
  H := Subgroup(G,[gen]);