/* Here we construct a subgroup H=PGL(2,13) inside 2F4(8). We first build a copy of PGL(2,13), which is generated
   by l1, l2=l2a*l2b, and h1. We then check that l1, l2 and h1 all lie in the standard magma copy of 2F4(8).
*/

q:=8;
F<w>:=GF(q);

l1:=GL(26,F)![[w^4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,w^2,0,0,0,0,0,0,0,0,w^3],
[0,w^3,w^4,w^2,w,0,0,0,0,0,0,0,0,0,0,0,0,w^6,1,w^5,w^4,w,w^2,1,w^6,0],
[0,w^3,w^5,w,0,0,0,0,0,0,0,0,0,0,0,0,0,w^6,w,w^4,0,w,w^3,w^6,0,0],
[0,w^3,w^5,w^4,w^3,0,0,0,0,0,0,0,0,0,0,0,0,w^6,w,1,w^6,w,w^3,w^2,w,0],
[0,w^3,w^4,0,w^2,0,0,0,0,0,0,0,0,0,0,0,0,w^6,1,0,w^5,w,w^2,0,1,0],
[0,1,w,w^6,w^5,w^5,w^6,w^4,w^3,0,0,0,0,0,0,0,0,w^3,w^4,w^2,w,w^6,1,w^5,w^4,0],
[0,1,w^2,w^5,0,w^5,1,w^3,0,0,0,0,0,0,0,0,0,w^3,w^5,w,0,w^6,w,w^4,0,0],
[0,1,w^2,w,1,w^5,1,w^6,w^5,0,0,0,0,0,0,0,0,w^3,w^5,w^4,w^3,w^6,w,1,w^6,0],
[0,1,w,0,w^6,w^5,w^6,0,w^4,0,0,0,0,0,0,0,0,w^3,w^4,0,w^2,w^6,1,0,w^5,0],
[w^6,0,0,0,0,0,0,0,0,w^5,w^2,0,0,0,0,0,w^4,0,0,0,0,0,0,0,0,w^2],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^2,w,w^5,w^4,1,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w,w,w^2,1,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^3,w,w^2,w^5,w,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^6,w,w^5,0,w^4,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,w^2,w^4,0,w^3,w^6,w^2,1,0,0,0,0,0,0,0,0,0,w],
[1,0,0,0,0,0,0,0,0,w^6,w^6,0,0,0,0,0,w^3,0,0,0,0,0,0,0,0,w^3],
[0,w^2,w^3,w,1,1,w,w^6,w^5,0,0,0,0,0,0,0,0,w,w^2,1,w^6,w,w^2,1,w^6,0],
[0,w^2,w^4,1,0,1,w^2,w^5,0,0,0,0,0,0,0,0,0,w,w^3,w^6,0,w,w^3,w^6,0,0],
[0,w^2,w^4,w^3,w^2,1,w^2,w,1,0,0,0,0,0,0,0,0,w,w^3,w^2,w,w,w^3,w^2,w,0],
[0,w^2,w^3,0,w,1,w,0,w^6,0,0,0,0,0,0,0,0,w,w^2,0,1,w,w^2,0,1,0],
[0,w^6,1,w^5,w^4,0,0,0,0,0,0,0,0,0,0,0,0,w^5,w^6,w^4,w^3,w^2,w^3,w,1,0],
[0,w^6,w,w^4,0,0,0,0,0,0,0,0,0,0,0,0,0,w^5,1,w^3,0,w^2,w^4,1,0,0],
[0,w^6,w,1,w^6,0,0,0,0,0,0,0,0,0,0,0,0,w^5,1,w^6,w^5,w^2,w^4,w^3,w^2,0],
[0,w^6,1,0,w^5,0,0,0,0,0,0,0,0,0,0,0,0,w^5,w^6,0,w^4,w^2,w^3,0,w,0],
[w^2,0,0,0,0,0,0,0,0,0,w,0,0,0,0,0,w^5,0,0,0,0,0,0,0,0,1]];

l2a:=GL(26,F)![[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w,w^4,w^4,w^3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^3,w^4,w^5,w^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^5,w^2,0,w^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^5,w^4,w^4,w^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w,w^4,w^4,w^3,w,w^4,w^4,w^3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^3,w^4,w^5,w^2,w^3,w^4,w^5,w^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^5,w^2,0,w^4,w^5,w^2,0,w^4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,w^5,w^4,w^4,w^2,w^5,w^4,w^4,w^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^4,w^4,w^2,w^2,w^5,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^6,w^5,w^2,w^6,w,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^4,w^6,w,0,w^2,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,w^6,w^6,w^2,w^2,w,0,0,0,0,0,0,0,0,0,0,0],
[w^5,0,0,0,0,0,0,0,0,1,0,w^3,w^3,w^2,w^2,1,0,0,0,0,0,0,0,0,0,0],
[w^2,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],
[0,w^5,w,w,1,w,w^4,w^4,w^3,0,0,0,0,0,0,0,0,w,w^4,w^4,w^3,0,0,0,0,0],
[0,1,w,w^2,w^6,w^3,w^4,w^5,w^2,0,0,0,0,0,0,0,0,w^3,w^4,w^5,w^2,0,0,0,0,0],
[0,w^2,w^6,0,w,w^5,w^2,0,w^4,0,0,0,0,0,0,0,0,w^5,w^2,0,w^4,0,0,0,0,0],
[0,w^2,w,w,w^6,w^5,w^4,w^4,w^2,0,0,0,0,0,0,0,0,w^5,w^4,w^4,w^2,0,0,0,0,0],
[0,w^4,1,1,w^6,w^6,w^2,w^2,w,0,0,0,0,0,0,0,0,w,w^4,w^4,w^3,w,w^4,w^4,w^3,0],
[0,w^6,1,w,w^5,w,w^2,w^3,1,0,0,0,0,0,0,0,0,w^3,w^4,w^5,w^2,w^3,w^4,w^5,w^2,0],
[0,w,w^5,0,1,w^3,1,0,w^2,0,0,0,0,0,0,0,0,w^5,w^2,0,w^4,w^5,w^2,0,w^4,0],
[0,w,1,1,w^5,w^3,w^2,w^2,1,0,0,0,0,0,0,0,0,w^5,w^4,w^4,w^2,w^5,w^4,w^4,w^2,0],
[w^5,0,0,0,0,0,0,0,0,w^6,w^4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1]];

l2b:=GL(26,F)![[0,w^5,w^5,w^6,w^4,w,0,w^4,w^5,0,0,0,0,0,0,0,0,w^5,w,w^6,w^5,1,w^5,0,w^2,0],
[w^6,w,1,0,w^3,1,w,w^3,0,w,w,w^2,w,w^2,w^6,0,w^4,w^3,w^2,w,w^4,w^5,w^5,w^4,w^6,w^2],
[w^3,w^4,0,w^6,w^4,w^2,w^2,w^2,w,1,w^3,w^6,w,1,w^2,0,w^6,w^5,w^4,w^5,w,w^3,w^5,w^5,w^2,0],
[w^5,w^6,0,w,w^6,w^2,w^2,w,w,w^2,w^2,w^6,w^6,1,w^3,0,1,w,w^4,w^5,w^5,0,w^5,1,w^5,w^5],
[w^2,w^5,w,w,0,w^5,1,w^6,w^5,w^4,w^2,w,w^6,w,w^2,0,0,0,0,w^5,w,w^4,1,w^4,1,1],
[w^2,1,1,w^5,0,w^4,w^6,0,w^4,w^6,w^4,w^2,w^6,w^3,w^2,0,w^6,w^5,w^6,0,w^4,0,w^6,0,0,w^3],
[w^6,w,0,w^6,w^2,w^4,w^4,w,w^3,w^2,w^3,w,w^6,w^4,w,0,w^4,w^2,w,w,w^4,w,1,w^5,0,w^6],
[w^3,w,w^5,w^3,w^5,w^6,w^5,w^6,1,w,1,0,w^2,w^2,w^5,0,w^2,w^5,w^4,w,w^4,w^2,w^6,0,w^4,w^6],
[1,w^6,w,1,w,w^3,w^6,w,w^6,w^4,w^6,w^6,w^2,1,w,0,w^5,0,w^4,0,w^5,1,w^4,w,1,w^4],
[w,0,w^5,w^4,w^3,w,w,w^5,1,w^3,1,w^2,w^6,w^2,w^5,1,1,w^5,w^6,w^3,w^3,0,1,w^6,w^4,0],
[w,0,0,0,0,0,0,0,0,w^3,1,w^2,w^6,w^2,w^5,1,1,0,0,0,0,0,0,0,0,0],
[w^6,w^5,1,w^6,w^4,w^6,w^3,w^5,w^6,w,w^5,1,w^4,1,w^3,w^5,w^5,w,w^6,1,w^4,1,1,w^2,w^5,0],
[w^3,w,1,w^5,w^6,0,w^6,w^3,1,w^5,w^2,w^4,w,w^4,1,w^2,w^2,w^3,w^5,w^4,0,1,0,w^6,w,0],
[1,w^6,w,w^4,w^5,w^5,w^4,w^3,w^2,w^2,w^6,w,w^5,w,w^4,w^6,w^6,w^2,1,w^5,0,w^6,w^2,w,w,0],
[w^3,w^4,w^3,w^2,w,w^3,w^2,w^5,w^2,w^5,w^2,w^4,w,w^4,1,w^2,w^2,w^6,w^6,w^4,w^6,w^5,w^3,w^2,1,0],
[0,w^2,w^3,w^6,1,w^2,w,1,w^2,0,1,0,0,0,0,0,0,w^5,w,w^4,w^3,w^6,w^6,w^4,0,0],
[w^4,w^5,w^2,w^6,w^6,w^5,0,w^6,w,w^6,w^3,w^5,w^2,w^5,w,w^3,w^3,w^5,w^2,0,0,w^5,w^3,1,w^4,0],
[w^5,w^3,w^4,w^4,w^6,w^6,0,w^3,w,w^6,w^4,w^3,0,w^2,w^6,0,0,1,w^2,w^3,w^4,w^6,w^4,w^6,0,w^6],
[w^6,1,0,1,w^6,0,w^6,w^4,w^6,0,1,0,1,w^6,w^2,0,w^5,w,1,0,0,w^2,1,w^3,w^2,w^3],
[0,0,w^3,1,w^3,w,w^5,w^4,w,w^3,w^3,w^2,w^5,1,w^5,0,1,w^3,w^3,w,0,w^6,w^2,w^5,w^4,w^4],
[w^4,w^4,w^5,w^2,w,w^6,w^2,w,w^4,w^5,w,w^6,w^4,w^5,w^6,0,w^6,w,w^3,w^4,w,1,0,w^6,w^3,w^3],
[w^2,0,0,w^3,w^4,w^5,w^6,w^5,w^5,w^3,w^2,1,w^3,w,1,0,1,w^5,w^6,1,w^5,w^6,0,w^3,w^3,w^6],
[w^2,w^4,0,w,0,w,w^5,w^2,w^2,w^4,0,w^3,0,w^3,w,0,w^5,w^5,w^4,w^6,w,w,w^5,w^5,w^6,w^6],
[1,w^3,w^5,w^4,w^3,1,w,w^4,0,0,w^6,w^2,w^2,w^6,w^4,0,0,0,w^6,w^6,w^3,w^3,w^2,w^6,w^2,0],
[w^5,1,0,w^6,w^4,w^2,w^4,w^5,w^4,w^2,1,w^3,w^6,w^2,w^4,0,1,w^5,0,w^2,w^2,w^4,w^2,w^4,w^5,w^6],
[w^2,w^5,w,w^2,w^5,w^4,w,w^3,w^3,w^4,w,w^3,1,w^3,w^6,w,w,w^6,w^3,0,w^6,w^6,1,w^4,w^5,0]];

l2:=l2a*l2b;

h1:=GL(26,F)![[w^3,w^2,w^2,1,0,0,w^2,w^3,w,0,1,w^3,w^2,0,w^4,0,0,w^4,w^6,w^6,w^3,1,w^5,0,w^2,0],
[w^5,0,w^4,w^2,0,w^3,w^3,w^5,w^3,0,w^2,w^5,1,0,w^6,w^5,w^5,1,w^2,1,w^5,w,w^2,1,w^2,w^2],
[w^4,w,w,w,0,w^4,w,w^4,w^2,w^4,w^3,w^6,0,w^5,w^5,0,w^6,w^2,w^5,w^3,w^2,w^6,w^3,w^5,1,0],
[w,w^5,w^4,0,0,w^3,w,w^2,w^2,w,w^3,1,w^6,w^4,1,w,w,w^4,w^3,w,w^5,w^2,w,w^3,w^2,w^5],
[w^6,w,w^2,1,w,w^2,w^5,w,w^2,1,0,0,1,w,1,w^3,w^5,0,w^3,w,w^2,1,w^2,w^6,w,1],
[0,w,w^3,w^6,0,w^2,w^5,w^5,w,w^4,w^6,w^6,1,w^3,1,w,0,w^5,0,0,w^4,w^2,w^5,w^2,w^5,w^3],
[w^3,w^3,w^5,w^6,w^2,w^3,1,w^4,w^4,1,w,w^4,w^6,w,w^4,w^2,0,0,w,w,0,w,w,w^3,1,w^6],
[1,1,w,w^5,w^5,w^5,w^2,w^6,w^6,w^5,w^6,0,w^5,w^2,w^6,w^4,0,w^3,w^3,w,0,w^3,w^3,w^5,w^2,w^6],
[0,w^3,w^6,w^5,1,w^2,1,0,1,0,w^4,w^3,w^4,w^4,w^2,w,0,0,w^3,0,w^5,0,w^4,w^2,1,w^4],
[w^3,w^5,1,w^5,w^3,1,1,w^6,w^3,0,0,w,w^6,1,w^5,1,1,0,0,0,0,w^5,w,w^6,w^5,0],
[0,1,1,1,0,w^5,w^5,1,w^4,w^3,w^4,w^6,0,w^2,w^6,1,1,w,w^4,w^2,w,w^3,w,0,w^5,0],
[w^5,w^6,w^5,w^3,w^4,w^5,w^2,0,1,0,w^4,w^2,w,w,w^5,w^6,w^5,w^2,w^6,w^4,1,1,1,w^5,w^6,w^4],
[w,0,w,w^6,1,1,w^6,w^5,w^3,0,1,w,w^3,w^4,w,w^2,1,w^4,w^2,w,w^3,w,w^4,w^5,0,0],
[w^6,w^4,0,w^3,w^5,w,1,w^4,1,0,0,1,0,w^3,w,0,w^6,w^4,w^5,w^6,1,1,w^6,0,1,w^2],
[w^5,0,w^3,w^3,w,w^4,w,w^2,w^6,w^6,0,w^5,1,w,w^2,w^6,w,w^3,0,w^4,w^6,0,1,w^6,w^5,w^3],
[w^3,w^6,w,w^3,w^2,0,w^5,w^5,1,w^3,w^3,0,0,1,w^4,w^4,0,w^4,w^6,w,w^6,0,w^3,w^3,w^2,1],
[w^2,w,w^2,1,w,w^4,0,w^3,w^4,w^6,w^3,w^6,0,0,0,w^3,0,0,w^5,1,w^4,1,w,w^4,0,0],
[0,w,1,0,0,w^6,w^3,w^4,w^3,w^4,1,w^6,1,w^3,1,w^4,w^3,1,w^6,w^4,w,w^2,w^2,w^2,w^3,w],
[w^6,0,0,w^4,w^5,w^3,w,w^3,w^4,w^3,w^5,w^2,w^4,w^5,0,1,w^6,0,w^6,w^4,w^5,w,w^2,w^4,w^5,w^3],
[w^2,w^2,w^5,1,w,1,1,w,w^3,0,w^5,w,1,w^6,w^2,w^5,1,1,w^2,1,w^5,w^5,w,w,w^3,w^2],
[w^4,w^3,0,w^4,w^3,w,1,w^3,w^6,w^4,0,w^4,w,1,w^5,w^5,1,w^2,w^5,w^3,w^2,w^2,w^3,w^4,w^3,0],
[0,w^2,0,0,0,w^3,w,w^5,0,w,w^2,w,w^5,1,w^4,0,w^3,1,w^5,w^2,0,w,0,0,0,0],
[0,w,0,0,0,w^4,1,w^4,0,1,w^3,w^3,w^3,w^6,w^3,1,w^5,w^5,w^5,w^6,w^6,1,0,w,w^2,1],
[0,0,w^4,0,0,0,w^5,0,1,w^2,w,w^3,0,w,w^5,1,1,w^6,w,w^5,w^3,w^2,w^4,w,w^4,w^2],
[w^3,w^4,0,w,w^2,w^3,w^2,0,w,w,w^6,0,w^4,0,w^6,1,w^5,w^3,1,w^3,w,w,w^5,w,0,w^2],
[0,w^3,0,0,0,w^4,w^2,w^6,0,w^2,w^3,w^5,w^6,w,w^5,0,w^3,0,1,w^3,0,w^6,w,w^4,w^5,w^3]];

H:=sub<GL(26,F)|l1,l2,h1>;
Bool:=IsIsomorphic(H,PGL(2,13)); Bool;
G:=LargeRee(8);
_:=RecogniseLargeRee(G);

LargeReeStandardMembership(l1) and LargeReeStandardMembership(l2) and LargeReeStandardMembership(h1);

