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:=3;; # 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> 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); 130 gap> GlobalParameters(gamma); [ [ 0, 0, 81 ], [ 1, 48, 32 ], [ 54, 27, 0 ] ] gap> Size(gamma.group); 12130560 gap> gap> Runtime(); 6850 gap> maxcliques:=CompleteSubgraphs(gamma);; gap> # gap> # maxcliques is now a generating set for the gamma.group-set of gap> # maximal cliques of gamma (which give the maximal partial spreads of gap> # PG(n,q)). gap> # gap> Runtime(); 7280 gap> gap> hom:=ActionHomomorphism(G,VertexNames(gamma),OnProjectivePointSets);; gap> GG:=Image(hom);; # GG = PGL(n+1,q) acting on the vertices of gamma. gap> # gap> # Now determine GG-orbit representatives of maxcliques. gap> # gap> reps:=List([1..Maximum(List(maxcliques,Size))],x->[]);; gap> for clique in maxcliques do > # determine whether clique represents a new orbit > k:=Length(clique); > equiv:=false; > for rep in reps[k] do > if RepresentativeAction(GG,clique,rep,OnSets)<>fail then > # clique and rep are in the same GG-orbit > equiv:=true; > break; > fi; > od; > if not equiv then > AddSet(reps[k],clique); > fi; > od; gap> Runtime(); 11380 gap> gap> maxreps:=Concatenation(reps);; gap> # gap> # Display the numbers of inequivalent maximal partial spreads of each size. gap> # gap> Collected(List(maxreps,Length)); [ [ 7, 1 ], [ 10, 2 ] ] gap> # gap> # Now display the representative maximal partial spreads. gap> # gap> for k in [1..Length(reps)] do > if Length(reps[k])<>0 then > Print("\n\nTHE ",Length(reps[k]), > " INEQUIVALENT MAXIMAL PARTIAL SPREAD(S) OF SIZE ", k,":\n"); > for rep in reps[k] do > Print("\nMaximal partial spread:\n",List(rep,x->List(VertexName(gamma,x), > v->List(v,y->readableFFE(F,y))))); > HH:=Stabilizer(GG,rep,OnSets); > H:=PreImage(hom,HH); > Print("\nThe stabilizer of this partial spread in PGL(",n+1,",",q,")", > " has size ",Size(HH)," and is generated by", > " the images of the matrices:\n"); > for g in GeneratorsOfGroup(H) do > Print(List(g,x->List(x,y->readableFFE(F,y))),",\n"); > od; > od; > fi; > od; THE 1 INEQUIVALENT MAXIMAL PARTIAL SPREAD(S) OF SIZE 7: Maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 0, 1 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 0 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 72 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 2, 2 ], [ 0, 2, 0, 2 ], [ 0, 0, 2, 1 ], [ 0, 0, 0, 2 ] ], [ [ 2, 1, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 0, 1, 0, 1 ], [ 1, 1, 1, 1 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 2 ] ], [ [ 1, 1, 1, 1 ], [ 0, 1, 0, 1 ], [ 1, 0, 1, 1 ], [ 2, 2, 1, 2 ] ], THE 2 INEQUIVALENT MAXIMAL PARTIAL SPREAD(S) OF SIZE 10: Maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ], [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 2, 0 ] ], [ [ 0, 1, 2, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 1, 0 ], [ 1, 2, 0, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 5760 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 2, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 2 ], [ 0, 0, 1, 1 ] ], [ [ 0, 2, 2, 0 ], [ 2, 0, 0, 1 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 1, 0, 0, 1 ], [ 0, 2, 1, 0 ], [ 0, 0, 2, 1 ], [ 0, 0, 1, 1 ] ], [ [ 2, 0, 1, 1 ], [ 0, 1, 1, 2 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 1, 0, 2 ], [ 2, 1, 1, 0 ], [ 0, 1, 0, 0 ], [ 2, 0, 0, 0 ] ], Maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 2 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ], [ [ 0, 1, 2, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 1, 0 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 2, 2 ], [ 1, 0, 1, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 2, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 1920 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 2, 0 ] ], [ [ 1, 2, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 2, 1 ] ], [ [ 1, 0, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 2, 2 ] ], [ [ 2, 2, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 0, 0, 2, 1 ], [ 0, 0, 2, 2 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ] ], [ [ 0, 2, 0, 1 ], [ 1, 0, 2, 0 ], [ 2, 2, 2, 2 ], [ 2, 1, 2, 1 ] ], gap> Runtime(); 12750 gap> # gap> # Now determine the (non-empty) non-maximal partial spreads gap> # from the maximal ones. gap> # gap> reps:=List([1..Length(reps)-1],x->[]);; gap> for maxrep in maxreps do > HH:=Stabilizer(GG,maxrep,OnSets); > for orb in Orbits(HH,Combinations(maxrep),OnSets) do > clique:=orb[1]; > k:=Length(clique); > if k>0 and k # Determine whether clique represents a new equivalence class > # of non-maximal partial spread. > equiv:=false; > for rep in reps[k] do > if RepresentativeAction(GG,clique,rep,OnSets)<>fail then > # clique and rep are equivalent > equiv:=true; > break; > fi; > od; > if not equiv then > AddSet(reps[k],clique); > fi; > fi; > od; > od; gap> Runtime(); 21260 gap> # gap> # Now display the representative non-maximal partial spreads gap> # (including the empty one). gap> # gap> for k in [0..Length(reps)] do > if k=0 then > sizekreps:=[[]]; > else > sizekreps:=reps[k]; > fi; > Print("\n\nTHE ",Length(sizekreps), > " INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE ", k,":\n"); > for rep in sizekreps do > Print("\nNon-maximal partial spread:\n", > List(rep,x->List(VertexName(gamma,x), > v->List(v,y->readableFFE(F,y))))); > HH:=Stabilizer(GG,rep,OnSets); > H:=PreImage(hom,HH); > Print("\nThe stabilizer of this partial spread in PGL(",n+1,",",q,")", > " has size ",Size(HH)," and is generated by", > " the images of the matrices:\n"); > for g in GeneratorsOfGroup(H) do > Print(List(g,x->List(x,y->readableFFE(F,y))),",\n"); > od; > od; > od; THE 1 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 0: Non-maximal partial spread: [ ] The stabilizer of this partial spread in PGL(4,3) has size 12130560 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 2, 0, 0, 1 ], [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ] ], THE 1 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 1: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 93312 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 1 ] ], [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 2 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 0, 0, 1 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ] ], [ [ 2, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 2, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 2, 1, 2, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 2, 1, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 0, 1 ] ], [ [ 0, 2, 0, 2 ], [ 2, 0, 1, 1 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 1 ] ], [ [ 2, 1, 1, 2 ], [ 2, 0, 2, 1 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 1 ] ], THE 1 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 2: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 2304 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 1 ] ], [ [ 2, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 2, 2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 0, 1, 2 ], [ 0, 0, 1, 0 ], [ 1, 1, 0, 0 ], [ 1, 0, 0, 0 ] ], THE 1 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 3: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 144 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 0, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 2, 1 ] ], [ [ 1, 0, 0, 0 ], [ 2, 2, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 2, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 1 ] ], [ [ 0, 1, 0, 1 ], [ 1, 2, 1, 2 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 1 ] ], [ [ 0, 1, 0, 1 ], [ 2, 2, 2, 2 ], [ 0, 2, 0, 0 ], [ 1, 1, 0, 0 ] ], THE 3 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 4: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 576 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 1, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 2, 1 ] ], [ [ 0, 2, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 1, 0, 0, 0 ], [ 2, 2, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 2, 2 ] ], [ [ 1, 0, 1, 0 ], [ 2, 1, 2, 1 ], [ 0, 0, 2, 0 ], [ 0, 0, 1, 2 ] ], [ [ 2, 0, 1, 0 ], [ 1, 2, 2, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 2, 1 ] ], [ [ 1, 0, 2, 0 ], [ 2, 1, 1, 2 ], [ 1, 0, 0, 0 ], [ 2, 1, 0, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 72 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 0, 2 ] ], [ [ 2, 0, 2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 0, 2, 1 ], [ 0, 2, 0, 1 ], [ 0, 0, 2, 2 ], [ 0, 0, 0, 1 ] ], [ [ 2, 1, 1, 1 ], [ 0, 2, 0, 1 ], [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 0, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 32 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 1 ] ], [ [ 0, 1, 0, 1 ], [ 2, 0, 2, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 1, 2, 0, 1 ], [ 0, 2, 2, 0 ], [ 2, 1, 0, 0 ], [ 0, 1, 0, 0 ] ], THE 4 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 5: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 32 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 1, 2, 0, 0 ], [ 2, 2, 0, 0 ], [ 0, 0, 2, 1 ], [ 0, 0, 1, 1 ] ], [ [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ], [ 1, 1, 0, 0 ], [ 2, 1, 0, 0 ] ], [ [ 2, 0, 1, 0 ], [ 0, 2, 0, 1 ], [ 2, 0, 2, 0 ], [ 0, 2, 0, 2 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 0, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 12 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 2, 1, 2 ], [ 0, 2, 0, 2 ], [ 0, 0, 2, 1 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 2, 1 ], [ 0, 2, 0, 1 ], [ 0, 0, 2, 2 ], [ 0, 0, 0, 1 ] ], [ [ 2, 1, 1, 1 ], [ 2, 2, 1, 0 ], [ 1, 1, 2, 1 ], [ 0, 1, 1, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 120 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 0, 2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 1, 0 ], [ 0, 1, 0, 2 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 2 ] ], [ [ 1, 2, 1, 1 ], [ 2, 2, 0, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 2, 1, 2, 2 ], [ 0, 1, 1, 2 ], [ 2, 1, 0, 0 ], [ 2, 2, 0, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 1, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 80 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [ [ 1, 0, 0, 2 ], [ 0, 2, 2, 0 ], [ 2, 2, 0, 0 ], [ 2, 1, 0, 0 ] ], THE 5 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 6: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 32 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 2, 0, 1 ], [ 1, 0, 2, 0 ] ], [ [ 1, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 1, 2, 2, 2 ], [ 2, 2, 2, 1 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 16 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 2, 1 ] ], [ [ 2, 2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 0, 0, 1, 0 ], [ 0, 0, 1, 2 ], [ 2, 0, 0, 0 ], [ 2, 1, 0, 0 ] ], [ [ 1, 0, 2, 0 ], [ 1, 2, 2, 1 ], [ 1, 0, 1, 0 ], [ 1, 2, 1, 2 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 192 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 1, 2, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 2 ], [ 0, 0, 1, 1 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 1, 2, 0, 0 ], [ 2, 2, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ], [ 2, 1, 0, 0 ], [ 2, 2, 0, 0 ] ], [ [ 1, 1, 2, 2 ], [ 2, 1, 1, 2 ], [ 2, 1, 2, 1 ], [ 2, 2, 2, 2 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 0, 1 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 36 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 2, 2 ], [ 0, 2, 0, 2 ], [ 0, 0, 2, 1 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 1, 0 ], [ 0, 1, 0, 2 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 2 ] ], [ [ 1, 2, 0, 1 ], [ 2, 2, 1, 2 ], [ 0, 1, 0, 0 ], [ 2, 2, 0, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 24 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 2, 2 ], [ 0, 2, 0, 2 ], [ 0, 0, 2, 1 ], [ 0, 0, 0, 2 ] ], [ [ 1, 2, 1, 1 ], [ 2, 2, 0, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 2, 2, 2 ], [ 2, 1, 0, 2 ], [ 1, 1, 2, 1 ] ], THE 3 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 7: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 1, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 48 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 0, 1, 1, 2 ], [ 2, 0, 1, 1 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 2 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 24 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 2 ] ], [ [ 2, 1, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 2 ] ], [ [ 0, 0, 1, 2 ], [ 0, 0, 0, 2 ], [ 1, 2, 0, 0 ], [ 0, 2, 0, 0 ] ], [ [ 2, 1, 1, 2 ], [ 0, 1, 0, 2 ], [ 1, 2, 1, 2 ], [ 0, 2, 0, 2 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 48 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 2, 1 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 0, 2, 1 ], [ 0, 0, 2, 2 ], [ 1, 1, 0, 0 ], [ 2, 1, 0, 0 ] ], [ [ 2, 1, 1, 2 ], [ 2, 2, 1, 1 ], [ 2, 2, 2, 2 ], [ 1, 2, 1, 2 ] ], THE 3 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 8: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 128 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 2, 2, 2, 1 ], [ 1, 2, 2, 2 ], [ 0, 0, 1, 1 ], [ 0, 0, 2, 1 ] ], [ [ 2, 1, 1, 1 ], [ 1, 1, 1, 2 ], [ 0, 0, 2, 1 ], [ 0, 0, 1, 1 ] ], [ [ 1, 2, 1, 1 ], [ 1, 1, 2, 1 ], [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 2, 2, 2 ], [ 1, 0, 1, 2 ], [ 1, 1, 1, 1 ], [ 2, 1, 2, 1 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 2 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 48 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 1 ] ], [ [ 2, 2, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 1, 2 ], [ 0, 0, 2, 2 ] ], [ [ 0, 0, 1, 2 ], [ 0, 0, 2, 2 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 1, 2, 2, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 2, 2 ], [ 2, 1, 2, 1 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ], [ [ 0, 1, 2, 2 ], [ 1, 0, 1, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 2, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 384 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 2, 1 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 1 ], [ 0, 0, 2, 2 ] ], [ [ 0, 2, 0, 0 ], [ 2, 1, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 2, 0 ] ], [ [ 0, 0, 1, 2 ], [ 0, 0, 0, 2 ], [ 2, 0, 0, 0 ], [ 1, 1, 0, 0 ] ], [ [ 1, 2, 2, 1 ], [ 0, 2, 0, 1 ], [ 1, 0, 1, 0 ], [ 2, 2, 2, 2 ] ], THE 2 INEQUIVALENT NON-MAXIMAL PARTIAL SPREAD(S) OF SIZE 9: Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 1, 2 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 1 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 2, 2 ], [ 1, 1, 0, 1 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ], [ [ 0, 1, 2, 1 ], [ 1, 0, 1, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 2, 0 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 576 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 0, 2, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 2 ], [ 0, 0, 1, 2 ] ], [ [ 0, 2, 2, 0 ], [ 2, 0, 0, 1 ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 0 ] ], [ [ 1, 2, 2, 1 ], [ 1, 1, 2, 2 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ] ], [ [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ], [ 0, 2, 0, 0 ], [ 2, 0, 0, 0 ] ], Non-maximal partial spread: [ [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ] ], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 1, 2, 0, 0 ] ], [ [ 0, 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 2 ], [ 1, 0, 2, 0 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 1 ] ], [ [ 0, 1, 1, 0 ], [ 1, 0, 0, 2 ], [ 1, 1, 1, 2 ], [ 1, 2, 2, 2 ] ], [ [ 0, 1, 1, 1 ], [ 1, 0, 2, 1 ], [ 1, 1, 0, 2 ], [ 1, 2, 1, 0 ] ], [ [ 0, 1, 1, 2 ], [ 1, 0, 1, 1 ], [ 1, 1, 2, 0 ], [ 1, 2, 0, 2 ] ], [ [ 0, 1, 2, 0 ], [ 1, 0, 0, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ] ], [ [ 0, 1, 2, 1 ], [ 1, 0, 2, 2 ], [ 1, 1, 1, 0 ], [ 1, 2, 0, 1 ] ] ] The stabilizer of this partial spread in PGL(4,3) has size 192 and is generated by the images of the matrices: [ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ], [ [ 2, 2, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 2, 1 ], [ 0, 0, 2, 2 ] ], [ [ 1, 1, 0, 0 ], [ 0, 2, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 2 ] ], [ [ 0, 0, 2, 0 ], [ 0, 0, 2, 1 ], [ 1, 1, 0, 0 ], [ 0, 2, 0, 0 ] ], [ [ 1, 0, 2, 0 ], [ 1, 2, 2, 1 ], [ 1, 1, 1, 1 ], [ 0, 2, 0, 2 ] ], gap> Runtime(); 23400 gap>