`Error, hash function not applicable to key of type data object` in `RecognizeGroup(SylowSubgroup(GL(6,2),3));`

[fingolfin]

Test case extracted from issue #13:

gap> RecognizeGroup(SylowSubgroup(GL(6,2),3));
Error, hash function not applicable to key of type data object in
  HTValue( orb!.ht, ob ) at GAPROOT/pkgorb/gap/orbits.gi:443 called from
Position( o, p ) at GAPROOT/pkggenss/gap/genss.gi:1308 called from
SiftGroupElementSLP( ri!.stabilizerchain, x ) at GAPROOT/pkg/recog/gap/projective.gi:109 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
...  at *stdin*:75
type 'quit;' to quit to outer loop
brk> Where(30);
Position( o, p ) at GAPROOT/pkggenss/gap/genss.gi:1308 called from
SiftGroupElementSLP( ri!.stabilizerchain, x ) at GAPROOT/pkg/recog/gap/projective.gi:109 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( riker, n ) at GAPROOT/pkg/recog/gap/base/recognition.gi:806 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:786 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( riker, n ) at GAPROOT/pkg/recog/gap/base/recognition.gi:806 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( riker, n ) at GAPROOT/pkg/recog/gap/base/recognition.gi:806 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:764 called from
SLPforElement( ImageRecogNode( ri ), ImageElm( Homom( ri ), x!.el ) ) at GAPROOT/pkg/recog/gap/base/kernel.gi:46 called from
GenerateRandomKernelElementsAndOptionallyVerifyThem( ri, n, false ) at GAPROOT/pkg/recog/gap/base/kernel.gi:91 called from
CallFuncList( findgensNmeth( ri ).method, Concatenation( [ ri ], findgensNmeth( ri ).args ) ) at GAPROOT/pkg/recog/gap/base/recognition.gi:635 called from
RecogniseGeneric( Group( List( GeneratorsOfGroup( H ), function ( x )
          return ImageElm( Homom( ri ), x );
      end ) ), methodsforimage( ri ), depthString, InitialDataForImageRecogNode( ri ) ) at GAPROOT/pkg/recog/gap/base/recognition.gi:608 called from
RecogniseGeneric( G, FindHomDbMatrix, "", rec(
     ) ) at GAPROOT/pkg/recog/gap/base/recognition.gi:112 called from
<function "RecogniseGroup">( <arguments> )
 called from read-eval loop at *errin*:2
brk> Grp(ri);
Group([ <an immutable 2x2 matrix over GF2>, <an immutable 2x2 matrix over GF2> ])
brk>

So we'd expects stuff over GF(2), but we see:

brk> ob;
[ 0*Z(2), Z(2^2) ]

The code in genss/gap/genss.gi:1308 is this:

InstallMethod( SiftGroupElementSLP, 
  "for a stabilizer chain and a group element",
  [ IsStabilizerChain and IsStabilizerChainByOrb, IsObject ],
  function( S, x )
    local preS, nrstrong, slp, o, p, po, r, isone;
    preS := false;
    isone := S!.IsOne;
    nrstrong := Length(S!.stronggens);
    slp := [];     # will be reversed in the end
    while S <> false do
        o := S!.orb;
        p := o!.op(o[1],x);
        po := Position(o,p);   # <- line 1308

Now, p is value ob we saw earlier. In fact, let's inspect some state:

brk> DownEnv();
brk> p;
[ 0*Z(2), Z(2^2)^2 ]
brk> o;
<closed orbit, 3 points with Schreier tree with log>
brk> o[1];
<a GF2 vector of length 2>
brk> Display(o[1]);
[ Z(2)^0, 0*Z(2) ]
brk> x;
[ [ 0*Z(2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2)^2 ] ]
brk> Display(x);
z = Z(4)
   . z^2
 z^2 z^2

So what is the action?

brk> Display(o!.op);
# src/listfunc.c:OnPoints
function ( pnt, elm )
    <<kernel code>> from src/listfunc.c:OnPoints
end

Hmmm... however...:

brk> GENSS_IsOneProjective = S!.IsOne;   # ah so we are acting projectively
true

So the action is linear, but the group is projective... and the matrix x is over GF(4) but it is a scalar multiple of a matrix in GF(2). So perhaps the action is supposed to be action?

Further evidence: the next caller up the chain is in gap/projective.gi:109 which is

SLPforElementFuncsProjective.StabilizerChainProj := function(ri,x)
  local r;
  r := SiftGroupElementSLP(ri!.stabilizerchain,x);  # <- line 109
  return r.slp;
end;

So where is the OnPoints coming from?? Hmm... from FindBasePointCandidates in genss :

    if q = 2 then
        op := OnPoints;
    else
        op := OnLines;
    fi;

ARGH. So yeah, sure over GF(2) there are no scalars other than 1, so we can use OnPoints as an optimization. But this is not a valid optimization if one allow scalar multiples of the matrices.

So I guess disabling this optimization might fix the issue. But I am not yet 100% sure if this is a proper fix, or more a cure of the symptom... Hm