gap> RequirePackage("grape");

Loading  GRAPE 4.0  (GRaph Algorithms using PErmutation groups),
by L.H.Soicher@qmw.ac.uk.

true
gap> 
gap> n:=3;; # projective dimension (>2)
gap> q:=7;; # field size
gap> 
gap> F:=GaloisField(q);;
gap> G:=GL(n+1,q);;
gap> zerovector:=ListWithIdenticalEntries(n+1,Zero(F));;
gap> p1:=ShallowCopy(zerovector);;
gap> p1[1]:=One(F);; 
gap> p2:=ShallowCopy(zerovector);;
gap> p2[2]:=One(F);;
gap> projectiveline:=Union([p2],List(Elements(F),x->p1+x*p2));;
gap> 
gap> readableFFE := function(F,x)
> #
> # Returns an object to print for the element x of the finite field F, 
> # for easy human reading when x is in the prime subfield of F. 
> #
> if x in PrimeField(F) then
>    return IntFFE(x);
> else
>    return x;
> fi;
> end;
function( F, x ) ... end
gap> 
gap> OnProjectivePointSets := function(projpointset,g)
> # 
> # returns the image of projective point set  projpointset  under the matrix  g.
> #
> return Set(List(projpointset,x->OnLines(x,g)));
> # Note: In GAP, a *line* is what we would call a representative of 
> # a projective point. 
> end;
function( projpointset, g ) ... end
gap> #
gap> # Now construct the G-graph gamma, whose vertices are labelled 
gap> # by the lines in PG(n,q), with two vertices joined by an edge 
gap> # iff the corresponding lines have no point in common. 
gap> #
gap> gamma:=Graph(G,
>              Set(Orbit(G,projectiveline,OnProjectivePointSets)),
>              OnProjectivePointSets,
>              function(x,y) 
>              # adjacency relation: join iff x and y are disjoint. 
>                 return Intersection(x,y)=[]; 
>              end,
>              true);;
gap> 
gap> # Display information about  gamma.
gap> OrderGraph(gamma);
2850
gap> GlobalParameters(gamma);
[ [ 0, 0, 2401 ], [ 1, 2016, 384 ], [ 2058, 343, 0 ] ]
gap> 
gap> hom:=ActionHomomorphism(G,VertexNames(gamma),OnProjectivePointSets);;
gap> GG:=Image(hom);; # GG = PGL(n+1,q) acting on the vertices of gamma.
gap> Runtime();
18800
gap> S:=SylowSubgroup(GG,5);;
gap> Size(S);
25
gap> IsCyclic(S);
true
gap> a:=First(List(GeneratorsOfGroup(S),x->x^5),y->Order(y)=5);;
gap> A:=Subgroup(GG,[a]);;
gap> Size(A);
5
gap> #
gap> # We shall classify maximal partial spreeads in PG(3,7) of size 45 
gap> # and invariant under A.
gap> #
gap> N:=Normalizer(GG,A);;
gap> Size(N);
1600
gap> alpha:=ShallowCopy(gamma);;
gap> Unbind(alpha.names);
gap> delta:=CollapsedCompleteOrbitsGraph(A,alpha,N);;
gap> OrderGraph(delta);
410
gap> Collected(List(VertexNames(delta),Length));
[ [ 5, 410 ] ]
gap> K:=CompleteSubgraphsOfGivenSize(delta,9);;
gap> Length(K);
320
gap> #
gap> # Now filter out the maximal cliques of  delta  that are in  K.
gap> #
gap> M:=Filtered(K,x->Intersection(List(x,y->Adjacency(delta,y)))=[]);;
gap> Length(M);
66
gap> #
gap> # Now determine  delta.group  orbit-representatives  of these maximal cliques.
gap> #
gap> M:=OrbitRepresentatives(delta.group,M,OnSets);;
gap> Length(M);
15
gap> #
gap> # Now create the corresponding cliques in  gamma.
gap> #
gap> K:=Set(List(M,x->Union(List(x,y->VertexName(delta,y)))));;
gap> Collected(List(K,Length));
[ [ 45, 15 ] ]
gap> #
gap> # Filter out the maximal cliques of  gamma  that are in  K.
gap> #
gap> M:=Filtered(K,x->Intersection(List(x,y->Adjacency(gamma,y)))=[]);;
gap> Length(M);
15
gap> Runtime();
150740
gap> stabilizers:=List(M,x->Stabilizer(GG,x,OnSets));;
gap> Runtime();
3124800
gap> Collected(List(stabilizers,IdGroup));
[ [ [ 10, 1 ], 9 ], [ [ 20, 3 ], 1 ], [ [ 60, 5 ], 1 ], [ [ 120, 34 ], 4 ] ]
gap> IdGroup(DihedralGroup(10));
[ 10, 1 ]
gap> IdGroup(AlternatingGroup(5));
[ 60, 5 ]
gap> IdGroup(SymmetricGroup(5));
[ 120, 34 ]
gap> #
gap> # In each stabilizer, all subgroups isomorphic to to  A
gap> # (i.e. subgroups of order 5) are conjugate, and so any equivalences 
gap> # amongst the cliques in M (under GG) can be achieved by 
gap> # an element of  N, the normalizer in GG of A,  and
gap> # all equivalences under  N  have already been taken care of 
gap> # (when we determined orbit-representatives with respect to 
gap> # delta.group).
gap> # 
gap> # Now display the representative maximal partial spreads of size 45
gap> # invariant under a group of order 5.
gap> #
gap> for i in [1..Length(M)] do
>    Print("\nMaximal partial spread:\n",List(M[i],x->List(VertexName(gamma,x),
>                                v->List(v,y->readableFFE(F,y)))));
>    Print("\nThe stabilizer of this partial spread in PGL(",n+1,",",q,")",
>          " has size ",Size(stabilizers[i])," and is generated by",
>          " the images of the matrices:\n");
>    for g in GeneratorsOfGroup(PreImage(hom,stabilizers[i])) do
>       Print(List(g,x->List(x,y->readableFFE(F,y))),",\n");
>    od;
> od; 

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 2 ], [ 1, 0, 3, 2 ], 
      [ 1, 0, 2, 2 ], [ 1, 0, 6, 2 ], [ 1, 0, 4, 2 ], [ 1, 0, 5, 2 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 1 ], [ 1, 1, 3, 3 ], 
      [ 1, 1, 2, 2 ], [ 1, 1, 6, 6 ], [ 1, 1, 4, 4 ], [ 1, 1, 5, 5 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 4, 0, 3 ], [ 1, 4, 1, 5 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 0 ], [ 1, 4, 6, 1 ], [ 1, 4, 4, 4 ], [ 1, 4, 5, 6 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 6, 0, 6 ], [ 1, 6, 1, 3 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 0 ], [ 1, 6, 6, 2 ], [ 1, 6, 4, 1 ], [ 1, 6, 5, 5 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 3, 0, 2 ], [ 1, 3, 1, 0 ], [ 1, 3, 3, 3 ], 
      [ 1, 3, 2, 5 ], [ 1, 3, 6, 4 ], [ 1, 3, 4, 1 ], [ 1, 3, 5, 6 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 6, 3 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 5, 3 ], [ 1, 4, 3, 3 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 1, 6 ], [ 1, 4, 6, 4 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 3, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 2, 5 ], [ 1, 4, 0, 1 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 4 ], [ 1, 1, 1, 3 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 6, 5 ], [ 1, 4, 4, 0 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 2, 3 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 0 ], [ 1, 4, 5, 4 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 1 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 1, 1 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 0, 1 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 4, 0 ], [ 1, 4, 5, 5 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 5, 4 ], [ 1, 1, 1, 6 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 2, 2 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 1, 0 ], [ 1, 1, 4, 6 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 5, 1 ], [ 1, 4, 6, 3 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 6, 4 ], [ 1, 1, 2, 1 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 4 ], [ 1, 1, 3, 5 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 6, 3 ], [ 1, 4, 2, 1 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 2 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 6, 5 ], [ 1, 1, 1, 0 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 3 ], [ 1, 4, 0, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 2, 1 ], [ 1, 1, 4, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 4 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 1 ], [ 1, 1, 6, 5 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 2, 4 ], [ 1, 4, 5, 3 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 4, 4 ], [ 1, 1, 3, 4 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 0, 4 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 3, 3 ], [ 1, 1, 2, 4 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 4, 2 ], [ 1, 4, 6, 0 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 5, 3 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 0, 4 ], [ 1, 4, 2, 5 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 0, 3 ], [ 1, 1, 6, 2 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 1, 4 ], [ 1, 4, 3, 6 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 2, 5 ], [ 1, 1, 1, 2 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 5, 0 ], [ 1, 1, 2, 0 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 0 ], [ 1, 4, 0, 0 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 3, 5 ], [ 1, 1, 0, 6 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 6, 4 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 0, 5 ], [ 1, 1, 4, 1 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 3 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 1, 6 ], [ 1, 1, 5, 1 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 4, 4 ], [ 1, 4, 3, 0 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 6, 1 ], [ 1, 1, 3, 6 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 2, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 0, 6 ], [ 1, 1, 5, 6 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 6 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 0, 5 ], [ 1, 4, 4, 3 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 6, 0 ], [ 1, 4, 3, 1 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 3, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 4 ], [ 1, 1, 0, 2 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 4, 6 ], [ 1, 4, 1, 3 ], [ 1, 5, 6, 1 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
60 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 6, 2, 5, 1 ], [ 0, 5, 0, 4 ], [ 5, 5, 1, 1 ], [ 0, 2, 0, 2 ] ],
[ [ 2, 1, 3, 2 ], [ 0, 2, 0, 1 ], [ 0, 2, 1, 2 ], [ 0, 0, 0, 1 ] ],
[ [ 2, 0, 3, 3 ], [ 4, 1, 4, 0 ], [ 6, 2, 1, 6 ], [ 1, 2, 2, 6 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 6 ], [ 1, 0, 1, 6 ], [ 1, 0, 3, 6 ], 
      [ 1, 0, 2, 6 ], [ 1, 0, 6, 6 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 6 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 3, 0, 6 ], [ 1, 3, 1, 0 ], [ 1, 3, 3, 2 ], 
      [ 1, 3, 2, 1 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 3 ], [ 1, 3, 5, 4 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 1, 0, 2 ], [ 1, 1, 1, 5 ], [ 1, 1, 3, 4 ], 
      [ 1, 1, 2, 1 ], [ 1, 1, 6, 6 ], [ 1, 1, 4, 0 ], [ 1, 1, 5, 3 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 3 ], [ 1, 6, 4, 6 ], [ 1, 6, 5, 1 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 6 ], [ 1, 2, 1, 5 ], [ 1, 2, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 2, 6, 0 ], [ 1, 2, 4, 2 ], [ 1, 2, 5, 1 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 5, 0, 1 ], [ 1, 5, 1, 5 ], [ 1, 5, 3, 6 ], 
      [ 1, 5, 2, 2 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 3 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 4, 0, 0 ], [ 1, 4, 1, 5 ], [ 1, 4, 3, 1 ], 
      [ 1, 4, 2, 3 ], [ 1, 4, 6, 2 ], [ 1, 4, 4, 6 ], [ 1, 4, 5, 4 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 5, 2 ], [ 1, 1, 6, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 4, 2 ], [ 1, 4, 2, 2 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 3, 3 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 1, 1 ], [ 1, 4, 6, 6 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 1, 0 ], [ 1, 1, 2, 3 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 0, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 6, 2 ], [ 1, 4, 4, 4 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 3, 4 ], [ 1, 1, 4, 1 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 2, 0 ], [ 1, 4, 0, 6 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 4, 1 ], [ 1, 1, 5, 6 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 1, 4 ], [ 1, 1, 4, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 4, 0 ], [ 1, 1, 0, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 5 ], [ 1, 4, 3, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 2, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 3, 5 ], [ 1, 1, 6, 4 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 6 ], [ 1, 4, 1, 1 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 2, 1 ], [ 1, 1, 5, 5 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 6, 4 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 0, 3 ], [ 1, 1, 3, 1 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 4, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 6, 1 ], [ 1, 1, 1, 1 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 4, 1 ], [ 1, 4, 0, 1 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 4, 4 ], [ 1, 1, 6, 5 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 2, 3 ], [ 1, 4, 5, 1 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 1, 3 ], [ 1, 1, 3, 6 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 6, 0 ], [ 1, 4, 2, 1 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 5, 4 ], [ 1, 1, 0, 3 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 0 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 3, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 1, 3 ], [ 1, 4, 4, 2 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 4, 5 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 5 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 3, 1 ], [ 1, 1, 2, 2 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 4, 0 ], [ 1, 4, 6, 5 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 0, 5 ], [ 1, 1, 6, 1 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 1, 2 ], [ 1, 4, 3, 3 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 6, 5 ], [ 1, 1, 5, 2 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 0, 1 ], [ 1, 4, 2, 0 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 4, 3 ], [ 1, 1, 1, 3 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 0, 3 ], [ 1, 4, 6, 3 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 3, 2 ], [ 1, 1, 0, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 6, 6 ], [ 1, 4, 5, 0 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 1, 2 ], [ 1, 1, 5, 1 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 4, 3 ], [ 1, 4, 3, 5 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 6, 4 ], [ 1, 1, 3, 2 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 2, 6 ], [ 1, 4, 1, 3 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 5, 0 ], [ 1, 1, 3, 0 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 0, 0 ], [ 1, 4, 4, 0 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 2, 5 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 4, 4 ], [ 1, 4, 1, 2 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 6, 3 ], [ 1, 1, 4, 2 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 1, 4 ], [ 1, 4, 5, 6 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 2, 6 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 6, 5 ], [ 1, 4, 3, 4 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 0, 2 ], [ 1, 1, 5, 0 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 2, 4 ], [ 1, 4, 6, 1 ], [ 1, 5, 4, 6 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 1, 2, 2, 3 ], [ 5, 3, 6, 3 ], [ 4, 5, 0, 5 ], [ 4, 1, 0, 3 ] ],
[ [ 2, 1, 3, 0 ], [ 0, 3, 2, 4 ], [ 3, 3, 2, 1 ], [ 2, 3, 1, 0 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 6 ], [ 1, 0, 1, 6 ], [ 1, 0, 3, 6 ], 
      [ 1, 0, 2, 6 ], [ 1, 0, 6, 6 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 6 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 4, 0, 6 ], [ 1, 4, 1, 0 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 1 ], [ 1, 4, 6, 5 ], [ 1, 4, 4, 3 ], [ 1, 4, 5, 4 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 5, 0, 0 ], [ 1, 5, 1, 3 ], [ 1, 5, 3, 2 ], 
      [ 1, 5, 2, 6 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 5 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 3, 0, 0 ], [ 1, 3, 1, 2 ], [ 1, 3, 3, 6 ], 
      [ 1, 3, 2, 4 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 1 ], [ 1, 3, 5, 3 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 6 ], [ 1, 2, 1, 5 ], [ 1, 2, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 2, 6, 0 ], [ 1, 2, 4, 2 ], [ 1, 2, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 5 ], [ 1, 1, 3, 1 ], 
      [ 1, 1, 2, 3 ], [ 1, 1, 6, 2 ], [ 1, 1, 4, 6 ], [ 1, 1, 5, 4 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 6, 2 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 3, 3 ], [ 1, 1, 4, 4 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 2, 2 ], [ 1, 4, 0, 0 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 5 ], [ 1, 1, 6, 1 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 4, 2 ], [ 1, 4, 2, 3 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 4 ], [ 1, 1, 2, 6 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 2 ], [ 1, 4, 5, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 2, 1 ], [ 1, 1, 3, 0 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 2 ], [ 1, 4, 6, 4 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 1 ], [ 1, 1, 5, 5 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 3, 4 ], [ 1, 4, 1, 3 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 6, 5 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 5, 0 ], [ 1, 4, 3, 4 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 4, 0 ], [ 1, 1, 0, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 5 ], [ 1, 4, 3, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 1, 5 ], [ 1, 1, 4, 0 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 5, 3 ], [ 1, 4, 6, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 3, 5 ], [ 1, 1, 6, 4 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 6 ], [ 1, 4, 1, 1 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 2, 2 ], [ 1, 1, 5, 6 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 6, 5 ], [ 1, 4, 0, 4 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 3, 1 ], [ 1, 1, 5, 1 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 1 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 4, 4 ], [ 1, 1, 6, 5 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 2, 3 ], [ 1, 4, 5, 1 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 3 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 6, 4 ], [ 1, 4, 2, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 0, 5 ], [ 1, 4, 3, 1 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 6, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 4, 1 ], [ 1, 4, 0, 3 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 5, 0 ], [ 1, 1, 0, 4 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 6, 2 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 4, 5 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 5 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 6, 2 ], [ 1, 1, 5, 3 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 0, 1 ], [ 1, 4, 2, 6 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 1, 3 ], [ 1, 1, 0, 2 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 2, 4 ], [ 1, 4, 4, 6 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 2, 4 ], [ 1, 1, 1, 1 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 0 ], [ 1, 4, 5, 6 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 0, 1 ], [ 1, 1, 6, 6 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 1, 3 ], [ 1, 4, 3, 0 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 4, 3 ], [ 1, 1, 1, 3 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 0, 3 ], [ 1, 4, 6, 3 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 6, 1 ], [ 1, 1, 3, 2 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 2, 0 ], [ 1, 4, 1, 5 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 3, 2 ], [ 1, 1, 0, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 6, 6 ], [ 1, 4, 5, 0 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 3 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 3, 5 ], [ 1, 4, 2, 0 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 5, 2 ], [ 1, 1, 2, 0 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 1 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 4 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 0, 4 ], [ 1, 4, 4, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 2, 5 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 4, 4 ], [ 1, 4, 1, 2 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 1, 0 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 5, 1 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 4, 2 ], [ 1, 1, 2, 1 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 6, 3 ], [ 1, 4, 3, 5 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 0, 5 ], [ 1, 1, 5, 2 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 1 ], [ 1, 4, 6, 0 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 6, 3 ], [ 1, 1, 4, 1 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 1, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 3, 0 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
120 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 1, 3, 0 ], [ 0, 5, 0, 0 ], [ 0, 5, 6, 0 ], [ 0, 0, 0, 6 ] ],
[ [ 1, 2, 4, 3 ], [ 2, 1, 0, 5 ], [ 3, 0, 1, 4 ], [ 4, 2, 4, 0 ] ],
[ [ 0, 1, 1, 0 ], [ 0, 0, 4, 5 ], [ 5, 0, 3, 2 ], [ 3, 1, 6, 4 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 6 ], [ 1, 0, 1, 6 ], [ 1, 0, 3, 6 ], 
      [ 1, 0, 2, 6 ], [ 1, 0, 6, 6 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 6 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 5, 0, 4 ], [ 1, 5, 1, 5 ], [ 1, 5, 3, 0 ], 
      [ 1, 5, 2, 6 ], [ 1, 5, 6, 3 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 4 ], [ 1, 2, 3, 3 ], 
      [ 1, 2, 2, 0 ], [ 1, 2, 6, 5 ], [ 1, 2, 4, 6 ], [ 1, 2, 5, 2 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 3 ], [ 1, 6, 4, 6 ], [ 1, 6, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 3, 0, 5 ], [ 1, 3, 1, 3 ], [ 1, 3, 3, 6 ], 
      [ 1, 3, 2, 1 ], [ 1, 3, 6, 0 ], [ 1, 3, 4, 4 ], [ 1, 3, 5, 2 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 5, 2 ], [ 1, 1, 6, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 4, 2 ], [ 1, 4, 2, 2 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 3, 2 ], [ 1, 1, 4, 3 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 2, 1 ], [ 1, 4, 0, 6 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 6, 3 ], [ 1, 1, 0, 6 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 5, 0 ], [ 1, 4, 3, 1 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 5 ], [ 1, 1, 3, 0 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 1, 3 ], [ 1, 4, 6, 6 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 6, 2 ], [ 1, 4, 4, 4 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 1, 5 ], [ 1, 1, 2, 2 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 0, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 4, 1 ], [ 1, 1, 5, 6 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 1, 4 ], [ 1, 1, 4, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 4, 0 ], [ 1, 1, 0, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 5 ], [ 1, 4, 3, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 2, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 0, 2 ], [ 1, 1, 3, 1 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 4, 3 ], [ 1, 4, 5, 5 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 5 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 0, 4 ], [ 1, 4, 1, 3 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 2, 4 ], [ 1, 1, 5, 2 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 6, 6 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 6, 1 ], [ 1, 1, 1, 1 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 4, 1 ], [ 1, 4, 0, 1 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 4, 2 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 3, 5 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 5, 0 ], [ 1, 1, 0, 2 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 1 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 1, 3 ], [ 1, 1, 3, 2 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 4 ], [ 1, 4, 2, 6 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 4 ], [ 1, 1, 6, 1 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 2, 0 ], [ 1, 4, 5, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 4, 5 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 5 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 0, 5 ], [ 1, 1, 6, 6 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 1, 4 ], [ 1, 4, 3, 2 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 6, 5 ], [ 1, 1, 5, 1 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 0, 2 ], [ 1, 4, 2, 3 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 1, 0 ], [ 1, 1, 0, 4 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 2, 3 ], [ 1, 4, 4, 2 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 3, 3 ], [ 1, 1, 2, 1 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 4, 5 ], [ 1, 4, 6, 2 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 4, 3 ], [ 1, 1, 1, 3 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 0, 3 ], [ 1, 4, 6, 3 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 3, 3 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 2, 4 ], [ 1, 4, 1, 5 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 4 ], [ 1, 1, 2, 3 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 1, 5 ], [ 1, 4, 0, 0 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 3, 5 ], [ 1, 1, 0, 3 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 6, 0 ], [ 1, 4, 5, 4 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 3, 4 ], [ 1, 1, 1, 5 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 5, 3 ], [ 1, 4, 2, 1 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 6, 4 ], [ 1, 1, 4, 0 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 1, 1 ], [ 1, 4, 5, 2 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 5, 5 ], [ 1, 1, 3, 4 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 1 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 2, 6 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 6, 5 ], [ 1, 4, 3, 4 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 0 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 4, 4 ], [ 1, 4, 1, 1 ], [ 1, 5, 6, 6 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 4, 5, 1, 1 ], [ 6, 3, 6, 1 ], [ 3, 4, 1, 1 ], [ 0, 6, 5, 2 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 4 ], [ 1, 0, 1, 4 ], [ 1, 0, 3, 4 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 4 ], [ 1, 0, 4, 4 ], [ 1, 0, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 6 ], [ 1, 6, 3, 1 ], 
      [ 1, 6, 2, 0 ], [ 1, 6, 6, 4 ], [ 1, 6, 4, 2 ], [ 1, 6, 5, 3 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 5, 0, 1 ], [ 1, 5, 1, 4 ], [ 1, 5, 3, 3 ], 
      [ 1, 5, 2, 0 ], [ 1, 5, 6, 5 ], [ 1, 5, 4, 6 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 6 ], 
      [ 1, 1, 2, 4 ], [ 1, 1, 6, 5 ], [ 1, 1, 4, 1 ], [ 1, 1, 5, 3 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 1 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 5 ], [ 1, 4, 6, 0 ], [ 1, 4, 4, 6 ], [ 1, 4, 5, 3 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 3, 0, 3 ], [ 1, 3, 1, 1 ], [ 1, 3, 3, 4 ], 
      [ 1, 3, 2, 6 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 2 ], [ 1, 3, 5, 0 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 3, 6 ], [ 1, 1, 4, 6 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 2, 6 ], [ 1, 4, 0, 6 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 6, 2 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 5, 1 ], [ 1, 4, 3, 6 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 3, 4 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 0 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 5 ], [ 1, 1, 1, 4 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 6, 6 ], [ 1, 4, 4, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 6, 0 ], [ 1, 1, 2, 0 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 5, 2 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 6, 0 ], [ 1, 4, 0, 5 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 0, 6 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 4, 3 ], [ 1, 4, 5, 4 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 1, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 5 ], [ 1, 4, 6, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 1, 1 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 3 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 0, 6 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 5 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 6 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 0, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 5 ], [ 1, 4, 2, 3 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 6, 3 ], [ 1, 1, 1, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 0 ], [ 1, 4, 0, 1 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 3, 0 ], [ 1, 1, 5, 6 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 3 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 5 ], [ 1, 1, 6, 2 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 2, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 6, 1 ], [ 1, 1, 5, 1 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 2, 1 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 3, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 3, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 4, 4 ], [ 1, 4, 6, 6 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 5, 4 ], [ 1, 4, 0, 3 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 1, 3 ], [ 1, 1, 0, 1 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 2, 5 ], [ 1, 4, 4, 2 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 3, 2 ], [ 1, 1, 0, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 6, 2 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 4, 4 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 0 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 1, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 6 ], [ 1, 4, 3, 0 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 5, 6 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 0 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 4, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 0, 0 ], [ 1, 4, 6, 2 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 3 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 6, 3 ], [ 1, 4, 3, 3 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 4 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 0, 1 ], [ 1, 1, 5, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 2 ], [ 1, 4, 6, 4 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 6 ], [ 1, 1, 0, 4 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 4, 1 ], [ 1, 4, 1, 5 ], [ 1, 5, 6, 3 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 0, 5, 0, 6 ], [ 4, 3, 6, 1 ], [ 1, 6, 6, 4 ], [ 3, 1, 2, 5 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 4 ], [ 1, 0, 1, 4 ], [ 1, 0, 3, 4 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 4 ], [ 1, 0, 4, 4 ], [ 1, 0, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 6 ], [ 1, 6, 3, 1 ], 
      [ 1, 6, 2, 0 ], [ 1, 6, 6, 4 ], [ 1, 6, 4, 2 ], [ 1, 6, 5, 3 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 2, 0, 3 ], [ 1, 2, 1, 5 ], [ 1, 2, 3, 2 ], 
      [ 1, 2, 2, 0 ], [ 1, 2, 6, 1 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 6 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 3 ], [ 1, 4, 3, 1 ], 
      [ 1, 4, 2, 2 ], [ 1, 4, 6, 5 ], [ 1, 4, 4, 0 ], [ 1, 4, 5, 6 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 1, 0, 5 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 3 ], 
      [ 1, 1, 2, 6 ], [ 1, 1, 6, 1 ], [ 1, 1, 4, 0 ], [ 1, 1, 5, 4 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 3, 0, 3 ], [ 1, 3, 1, 1 ], [ 1, 3, 3, 4 ], 
      [ 1, 3, 2, 6 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 2 ], [ 1, 3, 5, 0 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 6, 5 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 5 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 6, 2 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 5, 1 ], [ 1, 4, 3, 6 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 4, 3 ], [ 1, 1, 5, 6 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 3, 0 ], [ 1, 4, 1, 1 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 3, 4 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 0 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 6, 0 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 3, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 2, 2 ], [ 1, 4, 0, 1 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 0, 4 ], [ 1, 4, 5, 1 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 6, 3 ], [ 1, 1, 2, 3 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 3, 3 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 2 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 1, 5 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 1, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 5 ], [ 1, 4, 6, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 1, 1 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 3 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 0, 5 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 4, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 6, 6 ], [ 1, 1, 1, 6 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 4, 6 ], [ 1, 4, 0, 6 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 5 ], [ 1, 4, 2, 3 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 4, 1 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 3 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 3, 5 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 1, 3 ], [ 1, 4, 4, 6 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 5, 1 ], [ 1, 1, 0, 0 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 3, 2 ], [ 1, 4, 6, 4 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 6, 6 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 4 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 6, 1 ], [ 1, 1, 5, 1 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 2, 1 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 6, 4 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 1, 2 ], [ 1, 4, 3, 0 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 3, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 3, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 4, 4 ], [ 1, 4, 6, 6 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 5, 4 ], [ 1, 4, 0, 3 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 3, 2 ], [ 1, 1, 0, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 6, 2 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 6, 5 ], [ 1, 1, 3, 6 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 2, 4 ], [ 1, 4, 1, 2 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 0, 1 ], [ 1, 1, 4, 4 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 3, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 5, 6 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 0 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 5, 3 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 0 ], [ 1, 4, 3, 4 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 3, 0 ], [ 1, 1, 1, 0 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 4 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 0, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 0 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 6, 0 ], [ 1, 1, 4, 6 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 1, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 2, 4 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 6, 3 ], [ 1, 4, 3, 2 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 4, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 6, 5 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 0, 5, 0, 6 ], [ 4, 3, 6, 1 ], [ 1, 6, 6, 4 ], [ 3, 1, 2, 5 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 4 ], [ 1, 0, 1, 4 ], [ 1, 0, 3, 4 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 4 ], [ 1, 0, 4, 4 ], [ 1, 0, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 6 ], [ 1, 6, 3, 1 ], 
      [ 1, 6, 2, 0 ], [ 1, 6, 6, 4 ], [ 1, 6, 4, 2 ], [ 1, 6, 5, 3 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 4, 0, 6 ], [ 1, 4, 1, 1 ], [ 1, 4, 3, 5 ], 
      [ 1, 4, 2, 3 ], [ 1, 4, 6, 4 ], [ 1, 4, 4, 0 ], [ 1, 4, 5, 2 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 1, 0, 2 ], [ 1, 1, 1, 6 ], [ 1, 1, 3, 0 ], 
      [ 1, 1, 2, 3 ], [ 1, 1, 6, 5 ], [ 1, 1, 4, 4 ], [ 1, 1, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 3, 0, 3 ], [ 1, 3, 1, 1 ], [ 1, 3, 3, 4 ], 
      [ 1, 3, 2, 6 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 2 ], [ 1, 3, 5, 0 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 6 ], [ 1, 1, 2, 6 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 0, 6 ], [ 1, 4, 5, 6 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 6, 2 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 5, 1 ], [ 1, 4, 3, 6 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 3, 4 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 0 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 6, 0 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 3, 2 ], [ 1, 1, 4, 0 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 2, 4 ], [ 1, 4, 0, 1 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 5, 2 ], [ 1, 1, 1, 2 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 2, 2 ], [ 1, 4, 3, 2 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 2 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 1, 5 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 1, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 5 ], [ 1, 4, 6, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 6, 1 ], [ 1, 1, 2, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 2 ], [ 1, 4, 4, 4 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 0, 5 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 4, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 6, 0 ], [ 1, 1, 1, 1 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 6 ], [ 1, 4, 0, 4 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 4, 1 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 3 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 3, 5 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 1, 3 ], [ 1, 4, 4, 6 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 1, 2 ], [ 1, 1, 3, 1 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 6, 3 ], [ 1, 4, 2, 5 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 6, 6 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 4 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 6, 1 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 1, 1 ], [ 1, 4, 3, 1 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 6, 5 ], [ 1, 1, 5, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 0, 4 ], [ 1, 4, 2, 2 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 3, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 3, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 4, 4 ], [ 1, 4, 6, 6 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 5 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 5, 4 ], [ 1, 4, 0, 3 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 6, 3 ], [ 1, 1, 3, 3 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 2, 3 ], [ 1, 4, 1, 3 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 3, 6 ], [ 1, 1, 0, 0 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 3 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 0, 3 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 3, 0 ], [ 1, 4, 2, 1 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 5, 6 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 0 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 5, 3 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 0 ], [ 1, 4, 3, 4 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 3, 0 ], [ 1, 1, 1, 0 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 4 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 6, 2 ], [ 1, 4, 3, 0 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 0, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 0 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 5, 0 ], [ 1, 1, 3, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 0, 1 ], [ 1, 4, 4, 3 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 4, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 6, 5 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 0, 5, 0, 6 ], [ 4, 3, 6, 1 ], [ 1, 6, 6, 4 ], [ 3, 1, 2, 5 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 1, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 2 ], 
      [ 1, 1, 2, 2 ], [ 1, 1, 6, 2 ], [ 1, 1, 4, 2 ], [ 1, 1, 5, 2 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 3 ], [ 1, 0, 3, 5 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 1 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 0 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 0 ], [ 1, 4, 3, 6 ], 
      [ 1, 4, 2, 3 ], [ 1, 4, 6, 1 ], [ 1, 4, 4, 2 ], [ 1, 4, 5, 5 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 3 ], [ 1, 6, 4, 6 ], [ 1, 6, 5, 1 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 3, 0, 2 ], [ 1, 3, 1, 6 ], [ 1, 3, 3, 0 ], 
      [ 1, 3, 2, 3 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 4 ], [ 1, 3, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 5, 0, 2 ], [ 1, 5, 1, 0 ], [ 1, 5, 3, 3 ], 
      [ 1, 5, 2, 5 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 1, 5 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 0, 6 ], [ 1, 1, 1, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 6, 4 ], [ 1, 4, 4, 0 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 6, 4 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 5, 5 ], [ 1, 4, 3, 0 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 2, 3 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 0 ], [ 1, 4, 5, 4 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 1, 4 ], [ 1, 1, 4, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 3, 4 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 4, 2 ], [ 1, 4, 5, 0 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 5, 1 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 6, 2 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 2, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 4, 5 ], [ 1, 1, 0, 4 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 5 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 0, 4 ], [ 1, 4, 1, 3 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 5, 6 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 2, 1 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 6, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 6, 5 ], [ 1, 1, 1, 0 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 3 ], [ 1, 4, 0, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 2, 1 ], [ 1, 1, 4, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 4 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 4 ], [ 1, 1, 6, 1 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 2, 0 ], [ 1, 4, 5, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 0, 5 ], [ 1, 1, 6, 6 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 1, 4 ], [ 1, 4, 3, 2 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 2, 0 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 4, 1 ], [ 1, 4, 6, 2 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 1, 2 ], [ 1, 1, 0, 6 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 2, 5 ], [ 1, 4, 4, 4 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 6 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 5, 3 ], [ 1, 4, 0, 0 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 4, 2 ], [ 1, 1, 1, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 0, 1 ], [ 1, 4, 6, 6 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 3, 3 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 2, 4 ], [ 1, 4, 1, 5 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 3 ], [ 1, 4, 0, 5 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 3, 3 ], [ 1, 1, 0, 1 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 5 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 0, 3 ], [ 1, 4, 4, 1 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 6, 0 ], [ 1, 4, 3, 1 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 3, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 0 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 4, 4 ], [ 1, 4, 1, 1 ], [ 1, 5, 6, 6 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 4, 5, 1, 1 ], [ 6, 3, 6, 1 ], [ 3, 4, 1, 1 ], [ 0, 6, 5, 2 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 3, 0, 5 ], [ 1, 3, 1, 5 ], [ 1, 3, 3, 5 ], 
      [ 1, 3, 2, 5 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 5 ], [ 1, 3, 5, 5 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 3 ], [ 1, 0, 3, 5 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 1 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 0 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 2, 0, 6 ], [ 1, 2, 1, 2 ], [ 1, 2, 3, 1 ], 
      [ 1, 2, 2, 5 ], [ 1, 2, 6, 3 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 0 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 3 ], [ 1, 6, 4, 6 ], [ 1, 6, 5, 1 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 3 ], [ 1, 4, 3, 1 ], 
      [ 1, 4, 2, 2 ], [ 1, 4, 6, 5 ], [ 1, 4, 4, 0 ], [ 1, 4, 5, 6 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 1, 0, 5 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 3 ], 
      [ 1, 1, 2, 6 ], [ 1, 1, 6, 1 ], [ 1, 1, 4, 0 ], [ 1, 1, 5, 4 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 5, 0, 2 ], [ 1, 5, 1, 0 ], [ 1, 5, 3, 3 ], 
      [ 1, 5, 2, 5 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 1, 5 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 6, 6 ], [ 1, 1, 0, 2 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 5, 3 ], [ 1, 4, 3, 4 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 2 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 3, 5 ], [ 1, 4, 1, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 6, 0 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 5 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 4, 4 ], [ 1, 4, 2, 3 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 0, 4 ], [ 1, 4, 5, 1 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 1, 4 ], [ 1, 1, 4, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 2 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 1, 5 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 5, 1 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 6, 2 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 2, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 4, 5 ], [ 1, 1, 0, 4 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 0, 5 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 4, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 5, 6 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 2, 1 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 2, 2 ], [ 1, 1, 4, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 2 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 6, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 0 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 3, 3 ], [ 1, 4, 6, 6 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 6, 4 ], [ 1, 1, 1, 3 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 5 ], [ 1, 4, 0, 0 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 6, 6 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 4 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 6, 4 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 1, 2 ], [ 1, 4, 3, 0 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 3, 6 ], [ 1, 1, 2, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 4, 2 ], [ 1, 4, 6, 1 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 0, 3 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 2, 0 ], [ 1, 4, 4, 4 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 6, 5 ], [ 1, 1, 3, 6 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 2, 4 ], [ 1, 4, 1, 2 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 4, 4 ], [ 1, 1, 1, 0 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 0, 1 ], [ 1, 4, 6, 2 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 3 ], [ 1, 4, 0, 5 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 3, 3 ], [ 1, 1, 0, 1 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 5 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 0, 3 ], [ 1, 4, 4, 1 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 1, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 6, 4 ], [ 1, 4, 3, 6 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 3, 4 ], [ 1, 1, 1, 1 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 4, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 6, 5 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 0, 5, 0, 6 ], [ 4, 3, 6, 1 ], [ 1, 6, 6, 4 ], [ 3, 1, 2, 5 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 3, 0, 5 ], [ 1, 3, 1, 5 ], [ 1, 3, 3, 5 ], 
      [ 1, 3, 2, 5 ], [ 1, 3, 6, 5 ], [ 1, 3, 4, 5 ], [ 1, 3, 5, 5 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 3 ], [ 1, 0, 3, 5 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 1 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 0 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 2, 0, 6 ], [ 1, 2, 1, 2 ], [ 1, 2, 3, 1 ], 
      [ 1, 2, 2, 5 ], [ 1, 2, 6, 3 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 0 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 3 ], [ 1, 6, 4, 6 ], [ 1, 6, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 5, 0, 2 ], [ 1, 5, 1, 0 ], [ 1, 5, 3, 3 ], 
      [ 1, 5, 2, 5 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 6 ], [ 1, 1, 2, 6 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 0, 6 ], [ 1, 4, 5, 6 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 1, 5 ], [ 1, 4, 6, 3 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 6, 6 ], [ 1, 1, 0, 2 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 5, 3 ], [ 1, 4, 3, 4 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 2 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 3, 5 ], [ 1, 4, 1, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 6, 0 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 5 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 4, 4 ], [ 1, 4, 2, 3 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 3, 2 ], [ 1, 1, 4, 0 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 2, 4 ], [ 1, 4, 0, 1 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 1, 4 ], [ 1, 1, 4, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 5, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 2 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 1, 5 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 5, 1 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 6, 2 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 2, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 3 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 4, 5 ], [ 1, 1, 0, 4 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 6 ], [ 1, 4, 2, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 0, 5 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 4, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 5, 6 ], [ 1, 1, 1, 4 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 2, 1 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 2, 2 ], [ 1, 1, 4, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 2 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 6, 6 ], [ 1, 4, 2, 4 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 0 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 3, 3 ], [ 1, 4, 6, 6 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 6, 4 ], [ 1, 1, 1, 3 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 5 ], [ 1, 4, 0, 0 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 6, 6 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 4 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 6, 1 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 1, 1 ], [ 1, 4, 3, 1 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 6, 5 ], [ 1, 1, 5, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 0, 4 ], [ 1, 4, 2, 2 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 6 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 3 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 3, 6 ], [ 1, 1, 2, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 4, 2 ], [ 1, 4, 6, 1 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 0, 3 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 2, 0 ], [ 1, 4, 4, 4 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 6, 3 ], [ 1, 1, 3, 3 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 2, 3 ], [ 1, 4, 1, 3 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 4, 4 ], [ 1, 1, 1, 0 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 0, 1 ], [ 1, 4, 6, 2 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 6 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 3 ], [ 1, 4, 0, 5 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 3, 3 ], [ 1, 1, 0, 1 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 5 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 0, 3 ], [ 1, 4, 4, 1 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 1, 4 ], [ 1, 4, 5, 5 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 6, 4 ], [ 1, 4, 3, 6 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 3, 4 ], [ 1, 1, 1, 1 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 6 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 4, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 6, 5 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 5, 0, 5, 5 ], [ 0, 3, 0, 6 ], [ 3, 3, 2, 5 ], [ 0, 4, 0, 4 ] ],
[ [ 0, 5, 0, 6 ], [ 4, 3, 6, 1 ], [ 1, 6, 6, 4 ], [ 3, 1, 2, 5 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 6, 0, 4 ], [ 1, 6, 1, 4 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 4 ], [ 1, 6, 6, 4 ], [ 1, 6, 4, 4 ], [ 1, 6, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 3 ], [ 1, 0, 3, 5 ], 
      [ 1, 0, 2, 4 ], [ 1, 0, 6, 1 ], [ 1, 0, 4, 6 ], [ 1, 0, 5, 0 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 3, 0, 6 ], [ 1, 3, 1, 2 ], [ 1, 3, 3, 1 ], 
      [ 1, 3, 2, 5 ], [ 1, 3, 6, 3 ], [ 1, 3, 4, 4 ], [ 1, 3, 5, 0 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 6 ], 
      [ 1, 1, 2, 4 ], [ 1, 1, 6, 5 ], [ 1, 1, 4, 1 ], [ 1, 1, 5, 3 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 1 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 5 ], [ 1, 4, 6, 0 ], [ 1, 4, 4, 6 ], [ 1, 4, 5, 3 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 5, 0, 2 ], [ 1, 5, 1, 0 ], [ 1, 5, 3, 3 ], 
      [ 1, 5, 2, 5 ], [ 1, 5, 6, 4 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 3, 6 ], [ 1, 1, 4, 6 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 2, 6 ], [ 1, 4, 0, 6 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 4 ], [ 1, 1, 2, 6 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 2 ], [ 1, 4, 5, 5 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 5 ], [ 1, 1, 1, 4 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 6, 6 ], [ 1, 4, 4, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 6, 5 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 5, 0 ], [ 1, 4, 3, 4 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 6, 0 ], [ 1, 1, 2, 0 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 5, 1 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 6, 2 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 3, 2 ], [ 1, 0, 1, 5 ], [ 1, 1, 4, 0 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 5, 3 ], [ 1, 4, 6, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 0, 6 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 5 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 6 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 0, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 6, 3 ], [ 1, 1, 1, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 0 ], [ 1, 4, 0, 1 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 0, 5 ], [ 1, 4, 3, 1 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 3, 0 ], [ 1, 1, 5, 6 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 3 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 5 ], [ 1, 1, 6, 2 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 2, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 3, 2 ], [ 1, 1, 2, 2 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 4, 2 ], [ 1, 4, 6, 2 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 6 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 1, 6 ], [ 1, 1, 0, 2 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 2, 3 ], [ 1, 4, 4, 4 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 4, 4 ], [ 1, 1, 3, 3 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 0 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 0, 4 ], [ 1, 1, 6, 1 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 1, 0 ], [ 1, 4, 3, 6 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 6, 4 ], [ 1, 1, 5, 2 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 0, 6 ], [ 1, 4, 2, 3 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 4, 1 ], [ 1, 1, 1, 1 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 6, 1 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 4, 4 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 0 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 1, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 6 ], [ 1, 4, 3, 0 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 3, 4 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 2, 0 ], [ 1, 4, 1, 3 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 2, 1 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 3 ], [ 1, 4, 0, 5 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 3, 3 ], [ 1, 1, 0, 1 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 3 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 6, 3 ], [ 1, 4, 3, 3 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 1, 0 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 5, 1 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 0, 1 ], [ 1, 1, 5, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 2 ], [ 1, 4, 6, 4 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 6 ], [ 1, 1, 0, 4 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 4, 1 ], [ 1, 4, 1, 5 ], [ 1, 5, 6, 3 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
120 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 4, 2, 3, 6 ], [ 0, 1, 0, 3 ], [ 1, 1, 3, 6 ], [ 0, 6, 0, 6 ] ],
[ [ 5, 6, 4, 5 ], [ 0, 5, 0, 6 ], [ 0, 5, 6, 5 ], [ 0, 0, 0, 6 ] ],
[ [ 4, 0, 0, 5 ], [ 4, 3, 2, 2 ], [ 0, 0, 1, 6 ], [ 5, 2, 4, 6 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 0, 3 ], 
      [ 0, 1, 0, 2 ], [ 0, 1, 0, 6 ], [ 0, 1, 0, 4 ], [ 0, 1, 0, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 6, 0, 4 ], [ 1, 6, 1, 4 ], [ 1, 6, 3, 4 ], 
      [ 1, 6, 2, 4 ], [ 1, 6, 6, 4 ], [ 1, 6, 4, 4 ], [ 1, 6, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 3, 0, 3 ], [ 1, 3, 1, 4 ], [ 1, 3, 3, 6 ], 
      [ 1, 3, 2, 5 ], [ 1, 3, 6, 2 ], [ 1, 3, 4, 0 ], [ 1, 3, 5, 1 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 2 ], [ 1, 1, 3, 6 ], 
      [ 1, 1, 2, 4 ], [ 1, 1, 6, 5 ], [ 1, 1, 4, 1 ], [ 1, 1, 5, 3 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 1 ], [ 1, 2, 1, 0 ], [ 1, 2, 3, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 2 ], [ 1, 2, 4, 4 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 4, 0, 4 ], [ 1, 4, 1, 1 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 5 ], [ 1, 4, 6, 0 ], [ 1, 4, 4, 6 ], [ 1, 4, 5, 3 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 0, 0, 2 ], [ 1, 0, 1, 0 ], [ 1, 0, 3, 3 ], 
      [ 1, 0, 2, 5 ], [ 1, 0, 6, 4 ], [ 1, 0, 4, 1 ], [ 1, 0, 5, 6 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 6, 3 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 5, 3 ], [ 1, 4, 3, 3 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 4 ], [ 1, 1, 3, 5 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 3 ], [ 1, 4, 6, 1 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 2, 6 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 3, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 2, 5 ], [ 1, 4, 0, 1 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 0, 5 ], [ 1, 1, 1, 4 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 6, 6 ], [ 1, 4, 4, 1 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 0 ], [ 1, 1, 5, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 3, 3 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 2, 3 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 0 ], [ 1, 4, 5, 4 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 3, 0 ], [ 1, 0, 6, 0 ], [ 1, 1, 2, 0 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 3, 0 ], [ 1, 4, 4, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 0, 1 ], [ 1, 1, 3, 2 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 4, 0 ], [ 1, 4, 5, 5 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 1, 1 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 3 ], [ 1, 4, 3, 5 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 4, 2 ], [ 1, 1, 0, 6 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 1, 5 ], [ 1, 4, 2, 4 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 3, 5 ], [ 1, 0, 3, 1 ], [ 1, 1, 6, 6 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 0, 3 ], [ 1, 4, 1, 0 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 3, 5 ], [ 1, 4, 6, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 6 ], [ 1, 1, 3, 0 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 5 ], [ 1, 4, 2, 3 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 2 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 6, 5 ], [ 1, 1, 1, 0 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 3 ], [ 1, 4, 0, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 2, 1 ], [ 1, 1, 4, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 4 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 4, 5 ], [ 1, 1, 6, 2 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 2, 1 ], [ 1, 4, 5, 0 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 0 ], [ 1, 1, 2, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 5, 2 ], [ 1, 4, 1, 6 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 6, 1 ], [ 1, 1, 5, 1 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 2, 1 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 3, 1 ], [ 1, 4, 5, 6 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 3, 5 ], [ 1, 1, 2, 1 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 4, 2 ], [ 1, 4, 6, 3 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 3 ], [ 1, 1, 4, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 3, 0 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 4, 4 ], [ 1, 1, 3, 3 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 5, 5 ], [ 1, 4, 0, 0 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 0, 4 ], [ 1, 1, 6, 1 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 5, 5 ], [ 1, 6, 1, 0 ], [ 1, 4, 3, 6 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 3, 2 ], [ 1, 1, 0, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 6, 2 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 0, 3 ], [ 1, 1, 4, 4 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 3, 2 ], [ 1, 4, 2, 0 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 1, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 6 ], [ 1, 4, 3, 0 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 6, 2 ], [ 1, 1, 3, 4 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 2, 0 ], [ 1, 4, 1, 3 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 0 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 5, 6 ], [ 1, 4, 4, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 5, 4 ], [ 1, 1, 2, 2 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 6 ], [ 1, 4, 0, 3 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 0, 6 ], [ 1, 1, 5, 6 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 6 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 5, 0 ], [ 1, 1, 3, 1 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 4 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 6, 0 ], [ 1, 4, 3, 1 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 6, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 3, 6 ], [ 1, 4, 0, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 3, 6 ], [ 1, 1, 1, 5 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 6 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 1, 2 ], [ 1, 4, 5, 1 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 2, 6 ], [ 1, 1, 0, 4 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 4, 1 ], [ 1, 4, 1, 5 ], [ 1, 5, 6, 3 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
120 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 4, 2, 3, 6 ], [ 0, 1, 0, 3 ], [ 1, 1, 3, 6 ], [ 0, 6, 0, 6 ] ],
[ [ 5, 6, 4, 5 ], [ 0, 5, 0, 6 ], [ 0, 5, 6, 5 ], [ 0, 0, 0, 6 ] ],
[ [ 4, 0, 0, 5 ], [ 4, 3, 2, 2 ], [ 0, 0, 1, 6 ], [ 5, 2, 4, 6 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 3, 0 ], [ 0, 1, 3, 1 ], [ 0, 1, 3, 3 ], 
      [ 0, 1, 3, 2 ], [ 0, 1, 3, 6 ], [ 0, 1, 3, 4 ], [ 0, 1, 3, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 0, 0, 3 ], [ 1, 0, 1, 3 ], [ 1, 0, 3, 3 ], 
      [ 1, 0, 2, 3 ], [ 1, 0, 6, 3 ], [ 1, 0, 4, 3 ], [ 1, 0, 5, 3 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 4, 0, 1 ], [ 1, 4, 1, 2 ], [ 1, 4, 3, 4 ], 
      [ 1, 4, 2, 3 ], [ 1, 4, 6, 0 ], [ 1, 4, 4, 5 ], [ 1, 4, 5, 6 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 6, 0, 3 ], [ 1, 6, 1, 6 ], [ 1, 6, 3, 5 ], 
      [ 1, 6, 2, 2 ], [ 1, 6, 6, 0 ], [ 1, 6, 4, 1 ], [ 1, 6, 5, 4 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 3, 0, 5 ], [ 1, 3, 1, 0 ], [ 1, 3, 3, 4 ], 
      [ 1, 3, 2, 2 ], [ 1, 3, 6, 3 ], [ 1, 3, 4, 6 ], [ 1, 3, 5, 1 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 2, 0, 3 ], [ 1, 2, 1, 2 ], [ 1, 2, 3, 0 ], 
      [ 1, 2, 2, 1 ], [ 1, 2, 6, 4 ], [ 1, 2, 4, 6 ], [ 1, 2, 5, 5 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 5, 0, 6 ], [ 1, 5, 1, 3 ], [ 1, 5, 3, 4 ], 
      [ 1, 5, 2, 0 ], [ 1, 5, 6, 2 ], [ 1, 5, 4, 1 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 1, 0, 2 ], [ 1, 1, 1, 0 ], [ 1, 1, 3, 3 ], 
      [ 1, 1, 2, 5 ], [ 1, 1, 6, 4 ], [ 1, 1, 4, 1 ], [ 1, 1, 5, 6 ] ], 
  [ [ 0, 1, 0, 0 ], [ 1, 0, 3, 6 ], [ 1, 1, 3, 6 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 3, 6 ], [ 1, 4, 3, 6 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 1, 3 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 1, 1 ], [ 1, 4, 1, 6 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 0, 3 ], [ 1, 0, 4, 1 ], [ 1, 1, 4, 4 ], [ 1, 3, 4, 3 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 4, 5 ], [ 1, 4, 4, 6 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 5 ], [ 1, 1, 2, 0 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 2, 3 ], [ 1, 4, 2, 6 ], [ 1, 5, 2, 1 ] ], 
  [ [ 0, 1, 0, 6 ], [ 1, 0, 5, 2 ], [ 1, 1, 5, 1 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 5, 3 ], [ 1, 4, 5, 5 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 1, 0, 4 ], [ 1, 0, 0, 1 ], [ 1, 1, 0, 5 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 0, 4 ], [ 1, 4, 0, 3 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 0, 5 ], [ 1, 0, 6, 0 ], [ 1, 1, 6, 5 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 6, 2 ], [ 1, 4, 6, 6 ], [ 1, 5, 6, 4 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 3, 4, 1 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 0, 1 ], [ 1, 4, 5, 1 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 0, 4 ], [ 1, 1, 1, 5 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 6, 3 ], [ 1, 4, 4, 1 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 5, 6 ], [ 1, 1, 6, 2 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 4, 3 ], [ 1, 4, 2, 4 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 6, 1 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 5, 6 ], [ 1, 4, 3, 2 ], [ 1, 5, 4, 4 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 2, 6 ], [ 1, 1, 3, 5 ], [ 1, 3, 5, 3 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 1, 0 ], [ 1, 4, 6, 2 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 4, 5 ], [ 1, 1, 5, 2 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 3, 1 ], [ 1, 4, 1, 0 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 3, 5 ], [ 1, 1, 4, 3 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 2, 0 ], [ 1, 4, 0, 4 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 5, 4 ], [ 1, 1, 0, 4 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 3, 4 ], [ 1, 4, 6, 4 ], [ 1, 5, 1, 4 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 4, 6 ], [ 1, 1, 6, 0 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 3 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 2, 2 ], [ 1, 1, 4, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 0, 6 ], [ 1, 4, 3, 0 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 3, 1 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 1, 2 ], [ 1, 4, 4, 4 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 6, 4 ], [ 1, 1, 1, 1 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 4, 0 ], [ 1, 4, 0, 6 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 6 ], [ 1, 1, 2, 4 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 5, 1 ], [ 1, 4, 1, 5 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 6, 5 ], [ 1, 1, 5, 5 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 0, 5 ], [ 1, 4, 2, 5 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 3, 0 ], [ 1, 1, 2, 3 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 4, 4 ], [ 1, 4, 6, 5 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 6, 2 ], [ 1, 0, 5, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 3, 4 ], [ 1, 6, 6, 5 ], [ 1, 4, 1, 1 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 1, 0 ], [ 1, 1, 0, 6 ], [ 1, 3, 5, 4 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 2, 1 ], [ 1, 4, 4, 3 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 2, 1 ], [ 1, 1, 1, 6 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 3, 3 ], [ 1, 4, 5, 0 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 4, 0 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 3, 0 ], [ 1, 4, 2, 0 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 5, 5 ], [ 1, 1, 2, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 2 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 0, 0 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 6, 1 ], [ 1, 4, 5, 2 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 4, 2 ], [ 1, 1, 1, 4 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 0, 0 ], [ 1, 4, 6, 3 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 2, 4 ], [ 1, 1, 6, 3 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 5, 5 ], [ 1, 4, 4, 0 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 1, 6 ], [ 1, 1, 5, 3 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 4, 2 ], [ 1, 4, 3, 1 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 3, 2 ], [ 1, 1, 1, 2 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 5, 2 ], [ 1, 4, 2, 2 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 2, 0 ], [ 1, 1, 0, 1 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 4, 6 ], [ 1, 4, 1, 4 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 4 ], [ 1, 1, 6, 6 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 0, 5 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 0, 5 ], [ 1, 1, 5, 4 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 1 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 6, 2 ], [ 1, 1, 4, 6 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 1, 5 ], [ 1, 4, 5, 4 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 4, 4 ], [ 1, 1, 2, 2 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 6, 6 ], [ 1, 4, 3, 3 ], [ 1, 5, 1, 1 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
120 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 4, 3, 5, 3 ], [ 6, 5, 6, 0 ], [ 5, 5, 4, 4 ], [ 0, 6, 4, 1 ] ],
[ [ 6, 6, 3, 0 ], [ 1, 2, 3, 6 ], [ 2, 5, 2, 2 ], [ 0, 5, 1, 4 ] ],
[ [ 3, 5, 2, 3 ], [ 2, 3, 3, 3 ], [ 1, 1, 6, 3 ], [ 1, 4, 5, 6 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 0, 1, 3, 0 ], [ 0, 1, 3, 1 ], [ 0, 1, 3, 3 ], 
      [ 0, 1, 3, 2 ], [ 0, 1, 3, 6 ], [ 0, 1, 3, 4 ], [ 0, 1, 3, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 3, 0, 4 ], [ 1, 3, 1, 4 ], [ 1, 3, 3, 4 ], 
      [ 1, 3, 2, 4 ], [ 1, 3, 6, 4 ], [ 1, 3, 4, 4 ], [ 1, 3, 5, 4 ] ], 
  [ [ 0, 0, 1, 1 ], [ 1, 0, 0, 1 ], [ 1, 0, 1, 2 ], [ 1, 0, 3, 4 ], 
      [ 1, 0, 2, 3 ], [ 1, 0, 6, 0 ], [ 1, 0, 4, 5 ], [ 1, 0, 5, 6 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 2, 0, 2 ], [ 1, 2, 1, 5 ], [ 1, 2, 3, 4 ], 
      [ 1, 2, 2, 1 ], [ 1, 2, 6, 6 ], [ 1, 2, 4, 0 ], [ 1, 2, 5, 3 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 5, 0, 2 ], [ 1, 5, 1, 4 ], [ 1, 5, 3, 1 ], 
      [ 1, 5, 2, 6 ], [ 1, 5, 6, 0 ], [ 1, 5, 4, 3 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 6, 0, 1 ], [ 1, 6, 1, 0 ], [ 1, 6, 3, 5 ], 
      [ 1, 6, 2, 6 ], [ 1, 6, 6, 2 ], [ 1, 6, 4, 4 ], [ 1, 6, 5, 3 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 4 ], [ 1, 1, 3, 5 ], 
      [ 1, 1, 2, 1 ], [ 1, 1, 6, 3 ], [ 1, 1, 4, 2 ], [ 1, 1, 5, 6 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 4, 0, 1 ], [ 1, 4, 1, 6 ], [ 1, 4, 3, 2 ], 
      [ 1, 4, 2, 4 ], [ 1, 4, 6, 3 ], [ 1, 4, 4, 0 ], [ 1, 4, 5, 5 ] ], 
  [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 3 ], [ 1, 1, 0, 3 ], [ 1, 3, 0, 3 ], 
      [ 1, 2, 0, 3 ], [ 1, 6, 0, 3 ], [ 1, 4, 0, 3 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 0, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 2, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 2, 1 ], [ 1, 4, 2, 6 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 0, 3 ], [ 1, 0, 1, 5 ], [ 1, 1, 1, 1 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 1, 2 ], [ 1, 4, 1, 3 ], [ 1, 5, 1, 6 ] ], 
  [ [ 0, 1, 0, 2 ], [ 1, 0, 3, 6 ], [ 1, 1, 3, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 3, 4 ], [ 1, 4, 3, 0 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 0, 6 ], [ 1, 0, 6, 6 ], [ 1, 1, 6, 5 ], [ 1, 3, 6, 3 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 6, 0 ], [ 1, 4, 6, 2 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 1, 0 ], [ 1, 0, 2, 6 ], [ 1, 1, 3, 6 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 1, 6 ], [ 1, 4, 6, 6 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 0, 5 ], [ 1, 1, 1, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 6, 4 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 4, 4 ], [ 1, 1, 5, 0 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 6, 3 ], [ 1, 6, 3, 1 ], [ 1, 4, 1, 2 ], [ 1, 5, 2, 5 ] ], 
  [ [ 0, 1, 1, 2 ], [ 1, 0, 5, 2 ], [ 1, 1, 6, 4 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 4, 0 ], [ 1, 4, 2, 3 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 6, 1 ], [ 1, 1, 0, 5 ], [ 1, 3, 2, 6 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 5, 4 ], [ 1, 4, 3, 3 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 3, 1 ], [ 1, 1, 4, 6 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 2, 3 ], [ 1, 4, 0, 0 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 6, 2 ], [ 1, 1, 1, 2 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 2 ], [ 1, 4, 0, 2 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 2, 1 ], [ 1, 0, 4, 6 ], [ 1, 1, 6, 0 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 1, 1 ], [ 1, 6, 2, 5 ], [ 1, 4, 5, 3 ], [ 1, 5, 0, 4 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 2, 1 ], [ 1, 1, 4, 4 ], [ 1, 3, 1, 3 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 0, 5 ], [ 1, 4, 3, 6 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 3, 3 ], [ 1, 1, 5, 5 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 1, 1 ], [ 1, 4, 4, 4 ], [ 1, 5, 6, 6 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 4 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 3, 6 ], [ 1, 4, 6, 1 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 3 ], [ 1, 4, 2, 2 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 0, 6 ], [ 1, 1, 2, 4 ], [ 1, 3, 6, 0 ], 
      [ 1, 2, 4, 2 ], [ 1, 6, 5, 1 ], [ 1, 4, 1, 5 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 5, 1 ], [ 1, 1, 4, 1 ], [ 1, 3, 2, 1 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 6, 1 ], [ 1, 4, 1, 1 ], [ 1, 5, 0, 1 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 6, 3 ], [ 1, 1, 5, 4 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 0, 2 ], [ 1, 4, 2, 0 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 6, 3 ], [ 1, 0, 1, 3 ], [ 1, 1, 0, 6 ], [ 1, 3, 5, 5 ], 
      [ 1, 2, 6, 2 ], [ 1, 6, 2, 0 ], [ 1, 4, 4, 1 ], [ 1, 5, 3, 4 ] ], 
  [ [ 0, 1, 6, 6 ], [ 1, 0, 4, 1 ], [ 1, 1, 3, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 2, 6 ], [ 1, 6, 5, 2 ], [ 1, 4, 0, 4 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 6, 4 ], [ 1, 0, 3, 2 ], [ 1, 1, 2, 6 ], [ 1, 3, 0, 0 ], 
      [ 1, 2, 1, 3 ], [ 1, 6, 4, 5 ], [ 1, 4, 6, 4 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 6, 5 ], [ 1, 0, 2, 5 ], [ 1, 1, 1, 3 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 3, 0 ], [ 1, 4, 5, 4 ], [ 1, 5, 4, 2 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 4, 0 ], [ 1, 1, 1, 0 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 5, 0 ], [ 1, 6, 0, 0 ], [ 1, 4, 6, 0 ], [ 1, 5, 3, 0 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 0, 4 ], [ 1, 1, 4, 5 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 3, 3 ], [ 1, 4, 2, 1 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 4, 3 ], [ 1, 0, 1, 6 ], [ 1, 1, 5, 2 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 4, 3 ], [ 1, 4, 3, 4 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 3, 0 ], [ 1, 1, 0, 2 ], [ 1, 3, 1, 6 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 6, 5 ], [ 1, 4, 5, 1 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 6 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 2, 4 ], [ 1, 1, 6, 1 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 5, 0 ], [ 1, 4, 4, 6 ], [ 1, 5, 1, 3 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 3, 5 ], [ 1, 1, 1, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 6, 5 ], [ 1, 6, 5, 5 ], [ 1, 4, 2, 5 ], [ 1, 5, 0, 5 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 2, 0 ], [ 1, 1, 0, 1 ], [ 1, 3, 3, 3 ], 
      [ 1, 2, 5, 2 ], [ 1, 6, 4, 6 ], [ 1, 4, 1, 4 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 0, 0 ], [ 1, 1, 5, 3 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 3, 6 ], [ 1, 6, 2, 4 ], [ 1, 4, 6, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 1, 4 ], [ 1, 1, 6, 6 ], [ 1, 3, 2, 3 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 3, 2 ], [ 1, 4, 0, 5 ], [ 1, 5, 5, 0 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 6, 4 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 1, 5 ], [ 1, 4, 5, 0 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 5, 4 ], [ 1, 0, 4, 3 ], [ 1, 1, 2, 0 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 0, 4 ], [ 1, 6, 6, 6 ], [ 1, 4, 3, 5 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 2 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 3 ], [ 1, 5, 2, 1 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
10 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 2, 4, 0, 0 ], [ 5, 5, 0, 0 ], [ 3, 1, 1, 6 ], [ 0, 4, 5, 6 ] ],
[ [ 1, 2, 4, 5 ], [ 2, 0, 4, 6 ], [ 3, 3, 3, 2 ], [ 4, 4, 3, 0 ] ],

Maximal partial spread:
[ [ [ 0, 0, 0, 1 ], [ 1, 0, 4, 0 ], [ 1, 0, 4, 1 ], [ 1, 0, 4, 3 ], 
      [ 1, 0, 4, 2 ], [ 1, 0, 4, 6 ], [ 1, 0, 4, 4 ], [ 1, 0, 4, 5 ] ], 
  [ [ 0, 0, 1, 0 ], [ 1, 3, 0, 3 ], [ 1, 3, 1, 3 ], [ 1, 3, 3, 3 ], 
      [ 1, 3, 2, 3 ], [ 1, 3, 6, 3 ], [ 1, 3, 4, 3 ], [ 1, 3, 5, 3 ] ], 
  [ [ 0, 0, 1, 1 ], [ 0, 1, 0, 6 ], [ 0, 1, 1, 0 ], [ 0, 1, 3, 2 ], 
      [ 0, 1, 2, 1 ], [ 0, 1, 6, 5 ], [ 0, 1, 4, 3 ], [ 0, 1, 5, 4 ] ], 
  [ [ 0, 0, 1, 3 ], [ 1, 6, 0, 5 ], [ 1, 6, 1, 1 ], [ 1, 6, 3, 0 ], 
      [ 1, 6, 2, 4 ], [ 1, 6, 6, 2 ], [ 1, 6, 4, 3 ], [ 1, 6, 5, 6 ] ], 
  [ [ 0, 0, 1, 2 ], [ 1, 5, 0, 1 ], [ 1, 5, 1, 3 ], [ 1, 5, 3, 0 ], 
      [ 1, 5, 2, 5 ], [ 1, 5, 6, 6 ], [ 1, 5, 4, 2 ], [ 1, 5, 5, 4 ] ], 
  [ [ 0, 0, 1, 6 ], [ 1, 1, 0, 2 ], [ 1, 1, 1, 1 ], [ 1, 1, 3, 6 ], 
      [ 1, 1, 2, 0 ], [ 1, 1, 6, 3 ], [ 1, 1, 4, 5 ], [ 1, 1, 5, 4 ] ], 
  [ [ 0, 0, 1, 4 ], [ 1, 4, 0, 2 ], [ 1, 4, 1, 6 ], [ 1, 4, 3, 0 ], 
      [ 1, 4, 2, 3 ], [ 1, 4, 6, 5 ], [ 1, 4, 4, 4 ], [ 1, 4, 5, 1 ] ], 
  [ [ 0, 0, 1, 5 ], [ 1, 2, 0, 3 ], [ 1, 2, 1, 1 ], [ 1, 2, 3, 4 ], 
      [ 1, 2, 2, 6 ], [ 1, 2, 6, 5 ], [ 1, 2, 4, 2 ], [ 1, 2, 5, 0 ] ], 
  [ [ 0, 1, 0, 0 ], [ 1, 0, 1, 2 ], [ 1, 1, 1, 2 ], [ 1, 3, 1, 2 ], 
      [ 1, 2, 1, 2 ], [ 1, 6, 1, 2 ], [ 1, 4, 1, 2 ], [ 1, 5, 1, 2 ] ], 
  [ [ 0, 1, 0, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 2, 3 ], [ 1, 3, 2, 5 ], 
      [ 1, 2, 2, 4 ], [ 1, 6, 2, 1 ], [ 1, 4, 2, 6 ], [ 1, 5, 2, 0 ] ], 
  [ [ 0, 1, 0, 3 ], [ 1, 0, 3, 4 ], [ 1, 1, 3, 0 ], [ 1, 3, 3, 6 ], 
      [ 1, 2, 3, 3 ], [ 1, 6, 3, 1 ], [ 1, 4, 3, 2 ], [ 1, 5, 3, 5 ] ], 
  [ [ 0, 1, 0, 2 ], [ 1, 0, 0, 3 ], [ 1, 1, 0, 5 ], [ 1, 3, 0, 2 ], 
      [ 1, 2, 0, 0 ], [ 1, 6, 0, 1 ], [ 1, 4, 0, 4 ], [ 1, 5, 0, 6 ] ], 
  [ [ 0, 1, 0, 5 ], [ 1, 0, 6, 3 ], [ 1, 1, 6, 1 ], [ 1, 3, 6, 4 ], 
      [ 1, 2, 6, 6 ], [ 1, 6, 6, 5 ], [ 1, 4, 6, 2 ], [ 1, 5, 6, 0 ] ], 
  [ [ 0, 1, 1, 1 ], [ 1, 0, 0, 5 ], [ 1, 1, 1, 6 ], [ 1, 3, 3, 1 ], 
      [ 1, 2, 2, 0 ], [ 1, 6, 6, 4 ], [ 1, 4, 4, 2 ], [ 1, 5, 5, 3 ] ], 
  [ [ 0, 1, 1, 3 ], [ 1, 0, 6, 0 ], [ 1, 1, 0, 3 ], [ 1, 3, 2, 2 ], 
      [ 1, 2, 1, 6 ], [ 1, 6, 5, 4 ], [ 1, 4, 3, 5 ], [ 1, 5, 4, 1 ] ], 
  [ [ 0, 1, 1, 6 ], [ 1, 0, 5, 1 ], [ 1, 1, 6, 0 ], [ 1, 3, 1, 5 ], 
      [ 1, 2, 0, 6 ], [ 1, 6, 4, 2 ], [ 1, 4, 2, 4 ], [ 1, 5, 3, 3 ] ], 
  [ [ 0, 1, 1, 4 ], [ 1, 0, 2, 4 ], [ 1, 1, 3, 1 ], [ 1, 3, 5, 2 ], 
      [ 1, 2, 4, 5 ], [ 1, 6, 1, 0 ], [ 1, 4, 6, 6 ], [ 1, 5, 0, 3 ] ], 
  [ [ 0, 1, 1, 5 ], [ 1, 0, 3, 1 ], [ 1, 1, 4, 6 ], [ 1, 3, 6, 2 ], 
      [ 1, 2, 5, 4 ], [ 1, 6, 2, 3 ], [ 1, 4, 0, 0 ], [ 1, 5, 1, 5 ] ], 
  [ [ 0, 1, 3, 1 ], [ 1, 0, 1, 6 ], [ 1, 1, 4, 0 ], [ 1, 3, 3, 2 ], 
      [ 1, 2, 0, 1 ], [ 1, 6, 5, 5 ], [ 1, 4, 6, 3 ], [ 1, 5, 2, 4 ] ], 
  [ [ 0, 1, 3, 3 ], [ 1, 0, 3, 6 ], [ 1, 1, 6, 2 ], [ 1, 3, 5, 1 ], 
      [ 1, 2, 2, 5 ], [ 1, 6, 0, 3 ], [ 1, 4, 1, 4 ], [ 1, 5, 4, 0 ] ], 
  [ [ 0, 1, 3, 6 ], [ 1, 0, 2, 0 ], [ 1, 1, 5, 6 ], [ 1, 3, 4, 4 ], 
      [ 1, 2, 1, 5 ], [ 1, 6, 6, 1 ], [ 1, 4, 0, 3 ], [ 1, 5, 3, 2 ] ], 
  [ [ 0, 1, 3, 4 ], [ 1, 0, 5, 6 ], [ 1, 1, 1, 3 ], [ 1, 3, 0, 4 ], 
      [ 1, 2, 4, 0 ], [ 1, 6, 2, 2 ], [ 1, 4, 3, 1 ], [ 1, 5, 6, 5 ] ], 
  [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 3, 6, 1 ], 
      [ 1, 2, 4, 1 ], [ 1, 6, 5, 1 ], [ 1, 4, 1, 1 ], [ 1, 5, 3, 1 ] ], 
  [ [ 0, 1, 2, 3 ], [ 1, 0, 2, 5 ], [ 1, 1, 4, 1 ], [ 1, 3, 1, 0 ], 
      [ 1, 2, 6, 4 ], [ 1, 6, 0, 2 ], [ 1, 4, 3, 3 ], [ 1, 5, 5, 6 ] ], 
  [ [ 0, 1, 2, 2 ], [ 1, 0, 3, 5 ], [ 1, 1, 5, 0 ], [ 1, 3, 2, 4 ], 
      [ 1, 2, 0, 2 ], [ 1, 6, 1, 3 ], [ 1, 4, 4, 6 ], [ 1, 5, 6, 1 ] ], 
  [ [ 0, 1, 2, 6 ], [ 1, 0, 5, 5 ], [ 1, 1, 0, 4 ], [ 1, 3, 4, 2 ], 
      [ 1, 2, 2, 3 ], [ 1, 6, 3, 6 ], [ 1, 4, 6, 1 ], [ 1, 5, 1, 0 ] ], 
  [ [ 0, 1, 2, 4 ], [ 1, 0, 1, 0 ], [ 1, 1, 3, 4 ], [ 1, 3, 0, 5 ], 
      [ 1, 2, 5, 1 ], [ 1, 6, 6, 3 ], [ 1, 4, 2, 2 ], [ 1, 5, 4, 6 ] ], 
  [ [ 0, 1, 2, 5 ], [ 1, 0, 6, 6 ], [ 1, 1, 1, 4 ], [ 1, 3, 5, 0 ], 
      [ 1, 2, 3, 2 ], [ 1, 6, 4, 1 ], [ 1, 4, 0, 5 ], [ 1, 5, 2, 3 ] ], 
  [ [ 0, 1, 6, 0 ], [ 1, 0, 0, 6 ], [ 1, 1, 6, 6 ], [ 1, 3, 4, 6 ], 
      [ 1, 2, 5, 6 ], [ 1, 6, 1, 6 ], [ 1, 4, 3, 6 ], [ 1, 5, 2, 6 ] ], 
  [ [ 0, 1, 6, 1 ], [ 1, 0, 6, 1 ], [ 1, 1, 5, 2 ], [ 1, 3, 3, 4 ], 
      [ 1, 2, 4, 3 ], [ 1, 6, 0, 0 ], [ 1, 4, 2, 5 ], [ 1, 5, 1, 6 ] ], 
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      [ 1, 2, 6, 2 ], [ 1, 6, 2, 0 ], [ 1, 4, 4, 1 ], [ 1, 5, 3, 4 ] ], 
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      [ 1, 2, 3, 6 ], [ 1, 6, 6, 0 ], [ 1, 4, 1, 3 ], [ 1, 5, 0, 5 ] ], 
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      [ 1, 2, 1, 3 ], [ 1, 6, 4, 5 ], [ 1, 4, 6, 4 ], [ 1, 5, 5, 1 ] ], 
  [ [ 0, 1, 4, 0 ], [ 1, 0, 6, 5 ], [ 1, 1, 3, 5 ], [ 1, 3, 4, 5 ], 
      [ 1, 2, 0, 5 ], [ 1, 6, 2, 5 ], [ 1, 4, 1, 5 ], [ 1, 5, 5, 5 ] ], 
  [ [ 0, 1, 4, 1 ], [ 1, 0, 2, 3 ], [ 1, 1, 6, 4 ], [ 1, 3, 0, 6 ], 
      [ 1, 2, 3, 5 ], [ 1, 6, 5, 2 ], [ 1, 4, 4, 0 ], [ 1, 5, 1, 1 ] ], 
  [ [ 0, 1, 4, 2 ], [ 1, 0, 0, 0 ], [ 1, 1, 4, 2 ], [ 1, 3, 5, 6 ], 
      [ 1, 2, 1, 4 ], [ 1, 6, 3, 5 ], [ 1, 4, 2, 1 ], [ 1, 5, 6, 3 ] ], 
  [ [ 0, 1, 4, 6 ], [ 1, 0, 5, 3 ], [ 1, 1, 2, 2 ], [ 1, 3, 3, 0 ], 
      [ 1, 2, 6, 1 ], [ 1, 6, 1, 4 ], [ 1, 4, 0, 6 ], [ 1, 5, 4, 5 ] ], 
  [ [ 0, 1, 4, 4 ], [ 1, 0, 3, 3 ], [ 1, 1, 0, 0 ], [ 1, 3, 1, 1 ], 
      [ 1, 2, 4, 4 ], [ 1, 6, 6, 6 ], [ 1, 4, 5, 5 ], [ 1, 5, 2, 2 ] ], 
  [ [ 0, 1, 4, 5 ], [ 1, 0, 1, 5 ], [ 1, 1, 5, 3 ], [ 1, 3, 6, 6 ], 
      [ 1, 2, 2, 1 ], [ 1, 6, 4, 0 ], [ 1, 4, 3, 4 ], [ 1, 5, 0, 2 ] ], 
  [ [ 0, 1, 5, 0 ], [ 1, 0, 3, 0 ], [ 1, 1, 1, 0 ], [ 1, 3, 4, 0 ], 
      [ 1, 2, 6, 0 ], [ 1, 6, 5, 0 ], [ 1, 4, 2, 0 ], [ 1, 5, 0, 0 ] ], 
  [ [ 0, 1, 5, 1 ], [ 1, 0, 1, 4 ], [ 1, 1, 6, 5 ], [ 1, 3, 2, 0 ], 
      [ 1, 2, 4, 6 ], [ 1, 6, 3, 3 ], [ 1, 4, 0, 1 ], [ 1, 5, 5, 2 ] ], 
  [ [ 0, 1, 5, 3 ], [ 1, 0, 0, 2 ], [ 1, 1, 5, 5 ], [ 1, 3, 1, 4 ], 
      [ 1, 2, 3, 1 ], [ 1, 6, 2, 6 ], [ 1, 4, 6, 0 ], [ 1, 5, 4, 3 ] ], 
  [ [ 0, 1, 5, 2 ], [ 1, 0, 2, 6 ], [ 1, 1, 0, 1 ], [ 1, 3, 3, 5 ], 
      [ 1, 2, 5, 3 ], [ 1, 6, 4, 4 ], [ 1, 4, 1, 0 ], [ 1, 5, 6, 2 ] ], 
  [ [ 0, 1, 5, 6 ], [ 1, 0, 6, 4 ], [ 1, 1, 4, 3 ], [ 1, 3, 0, 1 ], 
      [ 1, 2, 2, 2 ], [ 1, 6, 1, 5 ], [ 1, 4, 5, 0 ], [ 1, 5, 3, 6 ] ], 
  [ [ 0, 1, 5, 5 ], [ 1, 0, 5, 4 ], [ 1, 1, 3, 2 ], [ 1, 3, 6, 5 ], 
      [ 1, 2, 1, 0 ], [ 1, 6, 0, 6 ], [ 1, 4, 4, 3 ], [ 1, 5, 2, 1 ] ] ]
The stabilizer of this partial spread in PGL(4,7) has size 
20 and is generated by the images of the matrices:
[ [ 5, 0, 0, 0 ], [ 0, 5, 0, 0 ], [ 0, 0, 5, 0 ], [ 0, 0, 0, 5 ] ],
[ [ 1, 3, 3, 5 ], [ 6, 0, 6, 4 ], [ 3, 6, 2, 4 ], [ 1, 0, 3, 4 ] ],
[ [ 6, 3, 4, 4 ], [ 3, 5, 2, 4 ], [ 4, 6, 4, 0 ], [ 1, 1, 1, 6 ] ],
gap> Runtime();
3151840
gap>