isomorphism test for (non-p-)groups: speed up?

[ThomasBreuer]

The following happens in GAP 4.16.0 and in the master branch.

We define two solvable groups of order $23^2 \cdot 48$ and ask whether they are isomorphic.

gap> G:=function()
> local g1,g2,g3,g4,g5,g6,g7,r,f,g,rws,x;
> f:=FreeGroup(IsSyllableWordsFamily,7);
> g:=GeneratorsOfGroup(f);
> g1:=g[1];
> g2:=g[2];
> g3:=g[3];
> g4:=g[4];
> g5:=g[5];
> g6:=g[6];
> g7:=g[7];
> rws:=SingleCollector(f,[ 2, 3, 2, 2, 2, 23, 23 ]);
> r:=[
> [1,g5],
> [3,g5],
> [4,g5],
> ];
> for x in r do SetPower(rws,x[1],x[2]);od;
> r:=[
> [2,1,g2],
> [3,1,g4],
> [4,1,g5],
> [6,1,g6^13*g7^18],
> [7,1,g6^21*g7^8],
> [3,2,g3*g4*g5],
> [4,2,g3],
> [6,2,g6^21*g7^11],
> [7,2,g6^2*g7^22],
> [4,3,g5],
> [6,3,g6^6*g7^18],
> [7,3,g6^10*g7^15],
> [6,4,g6^10*g7^9],
> [7,4,g6^12*g7^11],
> [6,5,g6^21],
> [7,5,g7^21],
> ];
> for x in r do SetCommutator(rws,x[1],x[2],x[3]);od;
> return GroupByRwsNC(rws);
> end;;
gap> G:=G();;
gap> H:=function()
> local g1,g2,g3,g4,g5,g6,g7,r,f,g,rws,x;
> f:=FreeGroup(IsSyllableWordsFamily,7);
> g:=GeneratorsOfGroup(f);
> g1:=g[1];
> g2:=g[2];
> g3:=g[3];
> g4:=g[4];
> g5:=g[5];
> g6:=g[6];
> g7:=g[7];
> rws:=SingleCollector(f,[ 2, 3, 2, 2, 2, 23, 23 ]);
> r:=[
> [1,g5],
> [3,g5],
> [4,g5],
> ];
> for x in r do SetPower(rws,x[1],x[2]);od;
> r:=[
> [2,1,g2],
> [3,1,g4],
> [4,1,g5],
> [6,1,g6^8*g7^5],
> [7,1,g6^2*g7^13],
> [3,2,g3*g4*g5],
> [4,2,g3],
> [6,2,g6^21*g7^11],
> [7,2,g6^2*g7^22],
> [4,3,g5],
> [6,3,g6^6*g7^18],
> [7,3,g6^10*g7^15],
> [6,4,g6^10*g7^9],
> [7,4,g6^12*g7^11],
> [6,5,g6^21],
> [7,5,g7^21],
> ];
> for x in r do SetCommutator(rws,x[1],x[2],x[3]);od;
> return GroupByRwsNC(rws);
> end;;
gap> H:=H();;
gap> IsomorphismGroups( G, H );

GAP runs for some time, growing to about 10 GB within 10 minutes.
(According to an experiment which I made yesterday, the computation will run and grow further.)

Interrupting the computation shows that IsomorphismGroups has called PatheticIsomorphism.
Looking at the code of this function, we find that another strategy (Morphium(G,H,false)) would have been chosen under different conditions, and that we can force this strategy by the global option forcetest:= "old".
For the given groups, this would help.

...
gap> G:= G();;
gap> H:= H();;
gap> IsomorphismGroups( G, H : forcetest:= "old" );
[ f1*f3*f5*f6^20*f7^8, f2*f6^8*f7^5 ] -> [ f1*f3*f6, f2 ]
gap> time;
619

Now the challenge is to find a condition that enables a good strategy for this example, without slowing down other examples.

[fingolfin]

I recently did run into a bunch of somewhat related issues. E.g.

order := 20412;
G:=PcGroupCode(10686578428610998262971791592184364557772754547145470755, order);
H:=PcGroupCode(5343819977472525260046739767108264858899801257522787107, order);
iso:=IsomorphismGroups( G, H );

I've aborted it after 40+ GB of RAM did not suffice.

I was able to settle that the two groups are not isomorphic with my own code that uses that they have a normal Sylow-7-subgroup, hence split into semidirect products; it then is a matter of verifying that the resulting "coupling" maps are in different orbits. But it'd be nice of course if IsomorphismGroups was able to handle these on its own.

[ThomasBreuer]

Concerning the groups of order 20412 mentioned above:
Here the nonisomorphism follows from the fact that the two character tables cannot be transformed into each other by permutations of rows and columns. In fact, already the matrices of irreducibles are already not permutation equivalent. (On the other hand, the set of character values, with multiplicities, is the same for the two groups.)