A spread of W(5,q) is a set of q3+1 planes which partition the points of W(5,q). Here we enumerate all spreads of W(5,3) which have a set-wise stabiliser of order a multiple of 13.
gap> w := SymplecticSpace(5,3);
W(5, 3)
gap> g := IsometryGroup(w);
#I Computing nice monomorphism...
PSp(6,3)
gap> syl := SylowSubgroup(g,13);
<projective group with Frobenius of size 13>
gap> planes := Planes(w);
<planes of W(5, 3)>
gap> points := Points(w);
<points of W(5, 3)>
gap> orbs := Orbits(syl,planes,OnPoints);;
gap> IsPartialSpread := x -> Number(points,p -> ForAny(x,i -> p in i)) = Size(x)*13;;
gap> 13s := Filtered(partialspreads,i -> Size(i)=13);;
gap> 26s := List(Combinations(13s,2),Union);;
gap> two := Union(Filtered(partialspreads,i -> Size(i)=1));;
gap> good26s := Filtered(26s,x->IsPartialSpread(Union(x,two)));;
gap> spreads := List(good26s,x -> Union(x,two));;
gap> spreads[1];
[ <a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)>, <a plane in W(5, 3)>, <a plane in W(5, 3)>,
<a plane in W(5, 3)> ]
