[GAP Forum] construction of an isomorphism between 2 different presentations

[Walter Becker]

This is in regard to a previous question to the GAP-forum 
regarding the identification of grooups being returned 
from the Group Construction program. See below for 
the program used:
 
LoadPackage("grpconst");
p:=11;
q:=5;
x:=ConstructAllGroups(q^3*p^2);;time;
LL:=Length(x);
i:=55;
H:=x[i];
ccs:=ConjugacyClasses(H);;
 classcounts := []; eltcounts := [];
for c in ccs do
 o := Order(Representative(c));
 if not IsBound(classcounts[o]) then classcounts[o] := 0; eltcounts[o] := 0;
fi;
 classcounts[o] := classcounts[o] + 1;
 eltcounts[o] := eltcounts[o] + Size(c);
 od;  
 for o in [1..Length(classcounts)] do
 if IsBound(classcounts[o]) then Print(o," ",eltcounts[o]," ",classcounts[o],"\n"); fi;
 od;
R:=PresentationViaCosetTable(H);
S:=FpGroupPresentation(R);
T:=RelatorsOfFpGroup(S);
Sq:=SylowSubgroup(H,q);
Rq:=PresentationViaCosetTable(Sq);
Sq:=FpGroupPresentation(Rq);
T3:=RelatorsOfFpGroup(Sq);
IdGroup(Sq);
 
 
The returned presentations here are
 
gap> T:=RelatorsOfFpGroup(S);
[ f2*f3*f2^-1*f3^-1, f2*f4*f2^-1*f4^-1, f3*f4*f3^-1*f4^-1, f1*f5*f1^-1*f5^-1,
  f2*f5*f2^-1*f5^-1, f3*f5*f3^-1*f5^-1, f4*f5*f4^-1*f5^-1,
  f5*f1*f2*f1^-1*f2^-1, f1^5, f1^-1*f3^3*f1*f3^-1, f1^-1*f4^3*f1*f4^-1,
  f1*f3^4*f1^-1*f3^-1, f1*f4^4*f1^-1*f4^-1, f1*f2^2*f1^-1*f2^3 ]
gap> Sq:=SylowSubgroup(H,q);
Group([ f1, f2, f5 ])
gap> Rq:=PresentationViaCosetTable(Sq);
<presentation with 3 gens and 5 rels of total length 25>
gap> Sq:=FpGroupPresentation(Rq);
<fp group on the generators [ f1, f2, f3 ]>
gap> T3:=RelatorsOfFpGroup(Sq);
[ f1*f3*f1^-1*f3^-1, f2*f3*f2^-1*f3^-1, f3*f1*f2*f1^-1*f2^-1, f1^5,
  f1*f2^2*f1^-1*f2^3 ]
gap> IdGroup(Sq);
[ 125, 4 ]
 
The question is to relate groups obtained from the Group 
Construction program as above to a presentation of  the form:
 
T_1^11=T_2^11=(T_1,T_2) = A^25=B^5=A^b*a^_6=
  T_1^A*T_1^a=T_2^A*T_2^b=T_1^B*T_1^c=T_2^B*T_2^d=1;
 
In this case I believe that  a and b are both -1 ie the order
11 group generators commute with the generator A, and the other 
5-group generator does not commute with the T's.
 
Question how to get the parameters c and d.   
 
Note there are several cases where the order structure of
the groups generated by  Grp Const program are (or appear to
be) the same as well as their automorphism groups.  Hence how
to assign the "constants a,b,c, and d to these groups obtained from
the Grp Const output.
 
In other  words how to assign  presentations of the above form (T_1,..B) to
groups in the Grp Const output.
 
Note the IsomorphismGroups(X,Y) may not work as the size of these
groups automorphism groups are quite large.
 
Suggestions ???
 
Walter Becker