/* In this case we let H denote a copy of PSL(2,13), L denote the subgroup 13.6, and let G denote E6(2^12).
   We give matrices defining L, an element of order 13 and an element of order 6. We construct l1pre and
   l2pre to lie in the normalizer of a torus NGT, as constructed using the GroupOfLieType command. We then
   give a matrix x1 to conjugate l1 and l2 to our versions, which we work with as the structure is nicer.
   We give H as L and a matrix h1, chosen at random. A posteriori, one could produce H as a subgroup of E6 and show that no other
   overgroup of L preserves the E6 trilinear form, but we prefer a less synthetic proof.

   We also give a matrix l3, which turns L into a 13.12, but does not normalize our subgroup H of G. This
   means that there are exactly two copies of H containing L.

   We also give a matrix j1 that, together with L, yields a copy of G2(3). We check that h1 lies in
   <l1,l2,j1>, and then that the two possible H<G are contained in H<J<G.
*/

q:=3^6;
F<zz>:=GF(q);
z:=zz^56;

l1pre:=GL(27,F)![[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],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],
[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],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],
[0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],
[0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],
[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],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0],
[0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],
[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],
[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]];

l2pre:=GL(27,F)!DiagonalMatrix([z^i:i in [3,0,4,11,12,10,7,11,8,5,2,9,1,6,6,5,3,12,2,0,9,10,0,1,8,7,4]]);

l3pre:=GL(27,F)![[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0],
[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],
[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
[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],
[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],
[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],
[0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0],
[0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0],
[0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0],
[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],
[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]];

x1:=GL(27,F)![[0,0,0,0,zz^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,zz^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,zz^726,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,zz^726,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,zz^726,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,zz^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,zz^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^726,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^726,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^2,0,0,0,0,0,0,0,0,0,0],
[0,zz^231,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^653,0,0,zz^194,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^726,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^2],
[0,zz^282,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^352,0,0,zz^627,0,0,0,0],
[zz^5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^724,0,0,0,0,0,0,0,0,0,0],
[0,zz^133,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^497,0,0,zz^497,0,0,0,0],
[0,0,zz^5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^360],
[0,0,0,zz^61,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,zz^61,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,zz^61,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,zz^61,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,zz^724,0,0,0,0,0,0,0,0,0,0,0,0,zz^5,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^61,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,zz^360,0,0,0,0,0,0,0,0,zz^5,0,0,0,0,0,0],
[0,0,0,0,0,zz^724,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^5,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,zz^360,0,0,0,0,0,0,0,0,0,0,zz^5,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^61,0]];

l1:=l1pre^(x1^-1);
l2:=l2pre^(x1^-1);
l3:=l3pre^(x1^-1);

L:=sub<GL(27,F)|l1,l2>;

h1:=GL(27,F)![[zz^588,zz^560,zz^238,zz^126,zz^154,zz^168,zz^420,zz^490,zz^238,zz^504,zz^447,zz^518,zz^672,0,zz^167,zz^83,zz^559,zz^615,zz^223,zz^111,zz^167,zz^727,zz^699,zz^475,zz^419,zz^335,zz^139],
[zz^560,zz^308,zz^154,zz^602,zz^126,zz^420,zz^504,zz^238,zz^154,zz^532,zz^83,zz^490,zz^588,zz^83,zz^475,0,zz^363,zz^139,zz^251,zz^699,zz^475,zz^419,zz^167,zz^335,zz^559,zz^531,zz^223],
[zz^378,zz^294,zz^504,zz^56,zz^588,zz^294,zz^266,zz^196,zz^672,zz^14,zz^153,zz^168,zz^630,zz^517,zz^349,0,zz^41,zz^41,zz^545,zz^629,zz^125,zz^601,zz^433,zz^97,zz^181,zz^13,zz^237],
[zz^266,zz^14,zz^56,zz^168,zz^308,zz^14,zz^658,zz^588,zz^560,zz^630,zz^153,zz^504,zz^294,zz^153,zz^13,zz^153,zz^545,zz^545,zz^601,zz^125,zz^69,zz^41,zz^265,zz^713,zz^237,zz^461,zz^405],
[zz^294,zz^266,zz^588,zz^308,zz^168,zz^658,zz^630,zz^56,zz^140,zz^378,zz^153,zz^196,zz^378,zz^153,zz^601,zz^153,zz^349,zz^265,zz^489,zz^237,zz^405,zz^461,zz^545,zz^181,zz^377,zz^41,zz^69],
[zz^168,zz^420,zz^154,zz^602,zz^518,zz^196,zz^672,zz^126,zz^602,zz^588,zz^447,zz^490,zz^140,zz^83,zz^727,zz^83,zz^167,zz^475,zz^531,zz^139,zz^251,zz^111,zz^223,zz^419,zz^699,zz^559,zz^335],
[zz^420,zz^504,zz^126,zz^518,zz^490,zz^672,zz^588,zz^602,zz^518,zz^560,zz^83,zz^238,zz^168,0,zz^419,zz^447,zz^475,zz^335,zz^111,zz^223,zz^503,zz^699,zz^251,zz^559,zz^167,zz^363,zz^531],
[zz^630,zz^378,zz^196,zz^588,zz^56,zz^266,zz^14,zz^504,zz^532,zz^658,zz^153,zz^308,zz^658,zz^517,zz^545,0,zz^461,zz^433,zz^265,zz^181,zz^237,zz^13,zz^41,zz^405,zz^713,zz^601,zz^125],
[zz^378,zz^294,zz^672,zz^560,zz^140,zz^14,zz^658,zz^532,zz^56,zz^630,zz^517,zz^224,zz^630,0,zz^405,0,zz^377,zz^125,zz^69,zz^41,zz^545,zz^713,zz^237,zz^601,zz^461,zz^181,zz^265],
[zz^504,zz^532,zz^602,zz^490,zz^238,zz^588,zz^560,zz^518,zz^490,zz^308,zz^447,zz^154,zz^420,zz^447,zz^559,0,zz^335,zz^531,zz^699,zz^251,zz^587,zz^167,zz^503,zz^363,zz^475,zz^55,zz^111],
[zz^645,zz^281,zz^211,zz^211,zz^211,zz^645,zz^281,zz^211,zz^575,zz^645,2,zz^211,zz^281,zz^252,1,zz^308,1,0,1,2,0,2,0,2,2,2,0],
[zz^658,zz^630,zz^168,zz^504,zz^196,zz^630,zz^378,zz^308,zz^224,zz^294,zz^153,zz^56,zz^14,0,zz^461,0,zz^601,zz^601,zz^41,zz^69,zz^629,zz^545,zz^489,zz^377,zz^405,zz^349,zz^181],
[zz^672,zz^588,zz^490,zz^154,zz^238,zz^140,zz^168,zz^518,zz^490,zz^420,zz^83,zz^602,zz^196,zz^447,zz^699,zz^447,zz^419,zz^587,zz^139,zz^531,zz^699,zz^195,zz^111,zz^167,zz^727,zz^475,zz^615],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,zz^392,2,0,1,2,1,1,0,1,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^308,2,zz^56,zz^196,zz^588,zz^196,zz^588,zz^504,zz^672,zz^224,zz^532,zz^196,zz^672],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,1,1,1,2,2,1,1,2,2,1,2],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^56,2,zz^196,zz^224,zz^672,zz^560,zz^224,zz^308,zz^308,zz^140,zz^224,zz^168,zz^196],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^196,2,zz^224,zz^168,zz^504,zz^560,zz^224,zz^560,zz^308,zz^672,zz^308,zz^588,zz^420],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^588,2,zz^672,zz^504,zz^56,zz^224,zz^672,zz^224,zz^196,zz^560,zz^196,zz^308,zz^532],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^196,1,zz^560,zz^560,zz^224,zz^56,zz^168,zz^672,zz^140,zz^672,zz^588,zz^588,zz^308],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^588,1,zz^224,zz^224,zz^672,zz^168,zz^504,zz^560,zz^420,zz^560,zz^308,zz^308,zz^196],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^504,2,zz^308,zz^560,zz^224,zz^672,zz^560,zz^588,zz^588,zz^532,zz^560,zz^56,zz^308],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^672,2,zz^308,zz^308,zz^196,zz^140,zz^420,zz^588,zz^168,zz^588,zz^560,zz^560,zz^224],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^224,1,zz^140,zz^672,zz^560,zz^672,zz^560,zz^532,zz^588,zz^196,zz^56,zz^672,zz^588],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^532,1,zz^224,zz^308,zz^196,zz^588,zz^308,zz^560,zz^560,zz^56,zz^308,zz^140,zz^224],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^196,2,zz^168,zz^588,zz^308,zz^588,zz^308,zz^56,zz^560,zz^672,zz^140,zz^588,zz^560],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,zz^672,1,zz^196,zz^420,zz^532,zz^308,zz^196,zz^308,zz^224,zz^588,zz^224,zz^560,zz^504]];

j1:=GL(27,F)![[zz^600,zz^613,zz^55,zz^301,zz^189,zz^681,zz^13,zz^211,zz^39,zz^108,zz^299,zz^506,zz^159,zz^462,zz^541,zz^42,zz^395,zz^662,zz^314,zz^613,zz^251,zz^486,zz^708,zz^618,zz^222,zz^90,zz^340],
[zz^167,zz^411,zz^240,zz^189,zz^431,zz^530,zz^435,zz^632,zz^174,zz^344,zz^395,zz^246,zz^193,zz^513,zz^292,zz^510,zz^331,zz^695,zz^329,zz^158,zz^347,zz^328,zz^520,zz^685,zz^450,zz^347,zz^313],
[zz^342,zz^488,zz^498,zz^173,zz^594,zz^642,zz^230,zz^109,zz^291,zz^6,zz^347,zz^560,zz^401,zz^633,zz^133,zz^589,zz^20,zz^305,zz^717,zz^186,zz^498,zz^630,zz^69,zz^462,zz^249,zz^79,zz^189],
[zz^626,zz^460,zz^672,zz^202,zz^221,zz^687,zz^127,zz^626,zz^322,zz^13,zz^513,zz^379,zz^239,zz^577,zz^197,zz^251,zz^499,zz^168,zz^439,zz^688,zz^585,zz^670,zz^262,zz^537,zz^262,zz^121,zz^453],
[zz^103,zz^37,zz^419,zz^386,zz^699,zz^214,zz^122,zz^78,zz^336,zz^526,zz^9,zz^416,zz^491,zz^111,zz^46,zz^430,zz^481,zz^625,zz^299,zz^284,zz^80,zz^93,zz^480,zz^446,zz^659,zz^638,zz^718],
[zz^409,zz^210,zz^130,zz^715,zz^606,zz^41,zz^332,zz^189,zz^71,zz^459,zz^392,zz^569,zz^351,zz^186,zz^727,zz^605,zz^301,zz^547,zz^101,zz^417,zz^237,zz^279,zz^616,zz^458,zz^633,zz^349,zz^207],
[zz^89,zz^297,zz^594,zz^430,zz^482,zz^675,zz^428,zz^323,zz^268,zz^110,zz^47,zz^351,zz^258,zz^533,zz^29,zz^415,zz^132,zz^452,zz^277,zz^31,zz^167,zz^200,zz^416,zz^374,zz^229,zz^702,zz^271],
[zz^370,zz^191,zz^462,zz^362,zz^644,zz^511,zz^323,zz^656,zz^630,zz^432,zz^291,zz^544,zz^374,zz^191,zz^676,zz^684,zz^186,zz^45,zz^361,zz^604,zz^621,zz^481,zz^600,zz^83,zz^304,zz^348,zz^594],
[zz^96,zz^270,zz^611,zz^607,zz^426,zz^681,zz^363,zz^112,zz^579,zz^230,zz^92,zz^286,zz^645,zz^166,zz^340,zz^123,zz^520,zz^399,zz^359,zz^410,zz^53,zz^425,zz^599,zz^644,zz^481,zz^342,zz^230],
[zz^666,zz^653,zz^180,zz^230,zz^142,zz^583,zz^682,zz^522,zz^510,zz^357,zz^367,zz^130,zz^325,zz^698,zz^57,zz^598,zz^185,zz^90,zz^466,zz^389,zz^664,zz^367,zz^507,zz^613,zz^39,zz^254,zz^501],
[zz^609,zz^301,zz^158,zz^24,zz^103,zz^354,zz^461,zz^405,zz^701,zz^3,zz^252,zz^624,zz^469,zz^350,zz^542,zz^447,zz^310,zz^103,zz^420,zz^520,zz^315,zz^162,zz^209,zz^3,1,zz^279,zz^366],
[zz^374,zz^550,zz^177,zz^224,zz^171,zz^634,zz^33,zz^218,zz^137,zz^214,zz^624,zz^70,zz^485,zz^354,zz^386,zz^148,zz^9,zz^574,zz^290,zz^465,zz^80,zz^224,zz^66,zz^320,zz^720,zz^581,zz^183],
[zz^389,zz^152,zz^268,zz^679,zz^326,zz^105,zz^387,zz^151,zz^142,zz^705,zz^476,zz^58,zz^195,zz^435,zz^94,zz^138,zz^284,zz^3,zz^196,zz^279,zz^504,zz^175,zz^479,zz^36,zz^369,zz^162,zz^537],
[zz^561,zz^253,zz^127,zz^463,zz^575,zz^533,zz^337,zz^155,zz^687,zz^617,2,zz^379,zz^421,zz^28,zz^362,zz^563,zz^35,zz^149,zz^154,zz^243,zz^75,zz^492,zz^92,zz^202,zz^55,zz^684,zz^479],
[zz^477,zz^309,zz^71,zz^211,zz^575,zz^533,zz^253,zz^603,zz^491,zz^197,zz^448,zz^211,zz^505,zz^532,zz^254,zz^348,zz^206,zz^623,zz^628,zz^180,zz^604,zz^473,zz^212,zz^147,zz^723,zz^73,zz^680],
[zz^673,zz^365,zz^323,zz^491,zz^267,zz^617,zz^197,zz^379,zz^351,zz^393,0,zz^547,zz^533,1,zz^204,zz^28,zz^693,zz^254,zz^231,zz^35,zz^380,zz^321,zz^30,zz^49,zz^176,zz^209,zz^324],
[zz^393,zz^169,zz^211,zz^155,zz^43,zz^309,zz^701,zz^155,zz^211,zz^57,zz^336,zz^127,zz^393,zz^504,zz^705,zz^726,zz^412,zz^572,zz^539,zz^628,zz^508,zz^218,zz^327,zz^85,zz^149,zz^23,zz^631],
[zz^533,zz^645,0,zz^99,zz^155,zz^477,zz^645,zz^491,zz^183,zz^141,zz^420,zz^687,zz^29,zz^224,zz^325,zz^619,zz^204,zz^188,zz^675,zz^650,zz^115,zz^181,zz^550,zz^652,zz^447,zz^264,zz^280],
[zz^281,zz^197,zz^127,0,zz^547,zz^281,zz^113,zz^323,zz^71,zz^589,zz^140,zz^659,zz,2,zz^237,zz^35,zz^161,zz^700,zz^505,zz^466,zz^462,zz^131,zz^515,zz^447,zz^573,zz^661,zz^459],
[zz^281,zz^645,zz^127,zz^687,zz^323,zz^197,zz^645,0,zz^659,zz^505,zz^336,zz^183,zz^225,zz^448,zz^60,zz^78,zz^569,zz^450,zz^171,zz^627,zz^560,zz^77,zz^553,zz^363,zz^456,zz^83,zz^480],
[zz^281,zz^505,zz^687,zz^183,0,zz^281,zz^477,zz^239,zz^127,zz^253,zz^112,zz^127,zz^645,zz^420,zz^103,zz^553,zz^92,zz^272,zz^539,zz^372,zz^374,zz^657,zz^308,zz^495,zz^425,zz^496,zz^291],
[zz^253,zz^253,zz^43,zz^687,zz^575,zz^337,zz,zz^435,zz^407,zz^449,zz^280,zz^211,zz^673,zz^476,zz^472,zz^381,zz^659,zz^450,zz^565,zz^253,zz^307,zz^536,zz^131,zz^681,zz^322,zz^20,zz^691],
[zz^113,zz^281,zz^547,zz^491,zz^99,zz^589,zz^29,zz^183,0,zz^645,zz^644,zz^323,zz^281,zz^700,zz^628,zz^481,zz^441,zz^262,zz^169,zz^588,zz^629,zz^463,zz^87,zz^365,zz^657,zz^376,zz^548],
[zz^225,zz^141,zz^575,zz^575,zz^323,zz^393,zz^309,zz^211,zz^659,zz^197,zz^252,zz^155,zz,zz^336,zz^381,zz^152,zz^388,zz^131,zz^385,zz^343,zz^259,zz^487,zz^607,zz^688,zz^137,zz^2,zz^702],
[zz,zz^197,zz^547,zz^575,zz^71,zz^421,zz^449,zz^211,zz^155,zz^169,zz^476,zz^71,zz^561,zz^84,zz^452,zz^533,zz^230,zz^261,zz^307,zz^572,zz^83,zz^649,zz^160,zz^530,zz^337,zz^435,zz^548],
[zz^169,zz^113,zz^211,zz^407,zz^99,zz^393,zz^617,zz^183,zz^211,zz^365,zz^392,zz^211,zz^225,zz^476,zz^296,zz^292,zz^585,zz^489,zz^576,zz^702,zz^55,zz^642,zz^590,zz^213,zz^395,zz^231,zz^83],
[zz^253,zz^309,zz^239,zz^547,zz^127,zz^225,zz^141,zz^407,zz^323,zz^645,zz^168,0,zz^281,zz^644,zz^495,zz^663,zz^628,zz^318,zz^112,zz^94,zz^438,zz^49,zz^386,zz^45,zz^120,zz^301,zz^592]];

J:=sub<GL(27,F)|l1,l2,j1>;

h1 in J;

// Here is the centralizer in GL(27,F) for L, even as a subgroup of NGT. This makes computation incredibly easy.

R<a,b,c,m1,m2,m3,m4,n1,n2,n3,n4>:=PolynomialRing(F,11);
mats:=MatrixRing(R,27);
scal:=mats!DiagonalMatrix([a:i in [1..27]]);
scal[2,20]:=b; scal[2,23]:=c; scal[20,2]:=c; scal[20,23]:=-b; scal[23,2]:=b; scal[23,20]:=-c;
a1:=[18,1,3,24,22,21];
a2:=[5,17,27,13,6,12];
a3:=[4,14,9,19,26,10];
a4:=[8,15,25,11,7,16];
for i in [1..#a1] do scal[a1[i],a1[i]]:=m1; scal[a1[i],a2[i]]:=m2; scal[a2[i],a1[i]]:=m3; scal[a2[i],a2[i]]:=m4; end for;
for i in [1..#a3] do scal[a3[i],a3[i]]:=n1; scal[a3[i],a4[i]]:=n2; scal[a4[i],a3[i]]:=n3; scal[a4[i],a4[i]]:=n4; end for;
scal[21,12]:=-m2;
scal[3,27]:=-m2;
scal[24,13]:=-m2;
scal[6,22]:=-m3;
scal[5,18]:=-m3;
scal[17,1]:=-m3;
scal[14,15]:=-n2;
scal[26,7]:=-n2;
scal[7,26]:=-n3;
scal[15,14]:=-n3;

scal*mats!l1pre eq mats!l1pre*scal and scal*mats!l2pre eq mats!l2pre*scal;

scal:=mats!x1*scal*mats!x1^-1;
// This commutes with l1x and l2x, but a change of basis gives us the below matrix. Which is nice.

scal:=mats![[m4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m1,0,0,0,0,0],
[0,m4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m1,0,0],
[0,0,n4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2*n3],
[0,0,0,n4,0,0,0,0,0,0,0,0,0,0,0,0,0,n3,0,0,0,0,0,0,0,0,0],
[0,0,0,0,n4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,n3,0,0,0,0],
[0,0,0,0,0,m4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2*m1,0,0,0],
[0,0,0,0,0,0,m4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2*m1,0],
[0,0,0,0,0,0,0,n4,0,0,0,0,0,0,0,0,0,0,0,0,2*n3,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,n4,0,0,0,0,0,0,0,0,0,0,n3,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,m4,0,0,0,0,m1,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,a,0,0,b,0,c,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,n4,0,0,0,0,0,0,n3,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,m4,0,0,0,2*m1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,a,0,b,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,2*m2,0,0,0,0,m3,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,a,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,m2,0,0,0,m3,0,0,0,0,0,0,0,0,0,0],
[0,0,0,n2,0,0,0,0,0,0,0,0,0,0,0,0,0,n1,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,n2,0,0,0,0,0,0,n1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,n2,0,0,0,0,0,0,0,0,0,0,n1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,2*n2,0,0,0,0,0,0,0,0,0,0,0,0,n1,0,0,0,0,0,0],
[2*m2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m3,0,0,0,0,0],
[0,0,0,0,n2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,n1,0,0,0,0],
[0,0,0,0,0,m2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m3,0,0,0],
[0,2*m2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m3,0,0],
[0,0,0,0,0,0,m2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,m3,0],
[0,0,2*n2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,n1]];

scal*mats!l1 eq mats!l1*scal and scal*mats!l2 eq mats!l2*scal;

seqs:=[[i,j,k]:i,j,k in [1..NumberOfRows(l1)]|i le j and j le k];

ents:=[39,43,78,152,155,157,204,241,271,286,311,321,359,475,560,563,565,581,601,619,626,640,669,706,763,821,848,851,853,1025,1039,
1046,1085,1088,1090,1196,1240,1271,1290,1330,1395,1422,1575,1617,1626,1656,1661,1729,1733,1737,1765,1767,1782,1804,1807,1844,1918,
1958,1973,1984,2018,2033,2043,2068,2207,2236,2283,2302,2354,2363,2381,2383,2429,2467,2496,2547,2592,2607,2617,2678,2689,2691,2730,
2734,2736,2758,2761,2780,2789,2795,2817,2822,2892,2918,2936,3000,3025,3032,3077,3097,3119,3141,3150,3156,3178,3183,3212,3222,3283,
3310,3319,3325,3347,3352,3372,3417,3453,3487,3525,3545,3593,3600];

vals:=[1,zz^358,zz^419,zz^192,zz^625,zz^495,zz^363,zz^550,zz^722,zz^255,zz^125,zz^352,zz^62,zz^727,zz^651,zz^350,zz^495,zz^645,zz^419,zz^722,
zz^344,zz^489,zz^352,zz^426,zz^716,zz^727,zz^279,zz^706,zz^123,zz^422,zz^413,zz^357,zz^548,zz^253,zz^123,zz^363,zz^419,zz^357,zz^49,zz^58,
zz^55,zz^352,zz^422,zz^413,zz^721,2,zz^358,zz^722,zz^593,zz^223,zz^644,zz^495,zz^274,zz^716,zz^125,zz^426,zz^55,zz^722,zz^186,zz^727,
zz^619,zz^716,zz^489,zz^426,zz^727,zz^357,zz^58,zz^413,zz^55,zz^585,zz^636,zz^487,zz^49,zz^58,zz^357,zz^645,zz^344,zz^426,zz^489,zz^352,
zz^377,zz^485,zz^223,zz^485,zz^393,zz^693,zz^393,zz^99,zz^405,zz^310,zz^347,zz^229,zz^58,zz^357,zz^413,zz^274,zz^125,zz^62,zz^716,zz^29,
zz^57,zz^257,zz^104,zz^15,zz^398,zz^682,zz^56,zz^680,zz^126,zz^680,zz^249,zz^613,zz^249,zz^316,zz^56,zz^126,zz^177,zz^420,zz^420,zz^541,
zz^56,zz^420];

f27:=[F!0:i in [1..#seqs]];
for i in [1..#ents] do f27[ents[i]]:=vals[i]; end for;

function Matricise(v)
return Matrix(1,NumberOfColumns(v),[R!v[i]:i in [1..NumberOfColumns(v)]]);
end function;

V:=GModule(L);
g:=mats!h1;

intseqs1:=[seqs[i]:i in ents];
intseqs:=&join{{[i[1],i[2],i[3]],[i[1],i[3],i[2]],[i[2],i[1],i[3]],[i[2],i[3],i[1]],[i[3],i[1],i[2]],[i[3],i[2],i[1]]}:i in intseqs1};

function ProdRel(seq)

u:=V.seq[1]; v:=V.seq[2]; w:=V.seq[3];
u1:=Matricise(u)*scal; v1:=Matricise(v)*scal; w1:=Matricise(w)*scal;
u2:=Matricise(u)*g*scal; v2:=Matricise(v)*g*scal; w2:=Matricise(w)*g*scal;
e1:=&+[u1[1,i[1]]*v1[1,i[2]]*w1[1,i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
e2:=&+[u2[1,i[1]]*v2[1,i[2]]*w2[1,i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];

return e1-e2;
end function;

function CheckRel(seq,gg)
u:=V.seq[1]; v:=V.seq[2]; w:=V.seq[3];
u2:=u^gg; v2:=v^gg; w2:=w^gg;
e1:=&+[u[i[1]]*v[i[2]]*w[i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
e2:=&+[u2[i[1]]*v2[i[2]]*w2[i[3]]*f27[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];
return e1-e2;
end function;

coeffs0:=[a,b,c,m1,m2,m3,m4,n1,n2,n3,n4];

function ChangeCoefficient(coeffs,coeff,target)
coeffs2:=coeffs;
nn:=Position(coeffs0,coeff);
coeffs2[nn]:=target;
return [Evaluate(i,coeffs2):i in coeffs];
end function;

function ChangeCoefficients(coeffs,coeff,target)
coeffs2:=coeffs;
for i in [1..#coeff] do
  nn:=Position(coeffs0,coeff[i]);
  coeffs2[nn]:=target[i];
end for;
return [Evaluate(i,coeffs2):i in coeffs];
end function;

function CheckDetermination()

ProdRel([1,2,13]) eq zz^237*(zz^499*a + zz^4*m2 + zz^46*m3 + zz^63*n1 - n2)^3;
c1:=ChangeCoefficient(coeffs0,n2,zz^499*a + zz^4*m2 + zz^46*m3 + zz^63*n1);

Evaluate(ProdRel([18,18,23]),c1) eq zz^393*(a+zz^600*m3)*(a+zz^233*m2+zz^591*m3)*(a+zz^597*m2+zz^639*m3);

// Kill off the two bad branches.

badc2a:=ChangeCoefficient(c1,m3,(a + zz^233*m2)/-zz^591);
badc2b:=ChangeCoefficient(c1,m3,-(a + zz^597*m2)/zz^639);

// Deal with badc2a first.
Evaluate(ProdRel([15,19,25]),badc2a) eq zz^537*(a+zz^558*m2)^3;
badc3a:=ChangeCoefficient(badc2a,m2,-a/zz^558);
Evaluate(ProdRel([15,16,16]),badc3a) eq zz^29*a^3;

// and branch b:

Evaluate(ProdRel([15,16,16]),badc2b) eq zz^321*a*n1*(a+zz^656*n1);
badc3b1:=ChangeCoefficient(badc2b,n1,0);
badc3b2:=ChangeCoefficient(badc2b,n1,-a/zz^656);

Evaluate(ProdRel([15,20,20]),badc3b1) eq zz^393*a^3;
Evaluate(ProdRel([15,20,20]),badc3b2) eq zz^393*a^3;

// On to the correct branch:

c2:=ChangeCoefficient(c1,m3,-a/zz^600);

Evaluate(ProdRel([15,18,23]),c2) eq zz^375*a*(a+zz^606*m2+zz^119*n1)*(a+zz^606*m2+zz^483*n1);

// Both of these turn out to be correct.

c3a:=ChangeCoefficient(c2,n1,(a+zz^606*m2)/-zz^119);
c3b:=ChangeCoefficient(c2,n1,(a+zz^606*m2)/-zz^483);

Evaluate(ProdRel([15,15,17]),c3a) eq zz^294*(a+zz^175*m2)^3;
Evaluate(ProdRel([15,15,17]),c3b) eq zz^469*(a+zz^420*m2)^3;
c4a:=ChangeCoefficient(c3a,m2,a/-zz^175);
c4b:=ChangeCoefficient(c3b,m2,a/-zz^420);

Evaluate(ProdRel([14,14,15]),c4a) eq zz^465*a^2*(a+zz^576*b);
Evaluate(ProdRel([14,14,15]),c4b) eq zz^569*a^2*(a+zz^576*b);

c5a:=ChangeCoefficient(c4a,b,a/-zz^576);
c5b:=ChangeCoefficient(c4b,b,a/-zz^576);

Evaluate(ProdRel([1,15,19]),c5a) eq zz^700*a^2*(a+zz^413*c+zz^489*n4);
Evaluate(ProdRel([1,19,25]),c5a) eq zz^299*a^2*(a+zz^541*c+zz^407*n3);
Evaluate(ProdRel([3,20,24]),c5a) eq zz^178*a^2*(a+zz^459*c+zz^248*m1);

Evaluate(ProdRel([1,15,19]),c5b) eq zz^702*a^2*(a+zz^229*c+zz^305*n4);
Evaluate(ProdRel([1,19,25]),c5b) eq zz^175*a^2*(a+zz^665*c+zz^713*n3);
Evaluate(ProdRel([3,20,24]),c5b) eq zz^461*a^2*(a+zz^358*c+zz^147*m1);

c6a:=ChangeCoefficients(c5a,[m1,n3,n4],[-(a+zz^459*c)/zz^248,-(a+zz^541*c)/zz^407,-(a+zz^413*c)/zz^489]);
c6b:=ChangeCoefficients(c5b,[m1,n3,n4],[-(a+zz^358*c)/zz^147,-(a+zz^665*c)/zz^713,-(a+zz^229*c)/zz^305]);

Evaluate(ProdRel([1,14,14]),c6a) eq zz^199*a^2*(a+zz^640*c+zz^368*m4);
Evaluate(ProdRel([1,14,14]),c6b) eq zz^390*a^2*(a+zz^126*c+zz^99*m4);

c7a:=ChangeCoefficient(c6a,m4,(a + zz^640*c)/-zz^368);
c7b:=ChangeCoefficient(c6b,m4,-(a + zz^126*c)/zz^99);

// This kills off everything.
c7a eq [a,zz^516*a,c,zz^116*a+zz^575*c,zz^189*a,zz^492*a,zz^724*a+zz^636*c,zz^709*a,zz^499*a,zz^685*a+zz^498*c,zz^603*a+zz^288*c];
c7b eq [a,zz^516*a,c,zz^217*a+zz^575*c,zz^672*a,zz^492*a,zz^265*a+zz^391*c,zz^527*a,zz^499*a,zz^379*a+zz^316*c,zz^59*a+zz^288*c];

// Remove the centralizer of H in GL(27,k):

c8a:=ChangeCoefficients(c7a,[a,c],[1,-1/zz^319]);
c8b:=ChangeCoefficients(c7b,[a,c],[1,-1/zz^128]);

sca1:=GL(27,F)!Evaluate(scal,c8a);
sca2:=GL(27,F)!Evaluate(scal,c8b);

h11:=h1^sca1;
h12:=h1^sca2;

H1:=sub<GL(27,F)|l1,l2,h11>;
H2:=sub<GL(27,F)|l1,l2,h12>;

H1^l3 eq H2;

j11:=j1^sca1;
j12:=j1^sca2;

// This gives us exactly two groups H containing the group L, contained in two different groups J.

{CheckRel(i,j11):i in seqs} eq {0};
{CheckRel(i,j12):i in seqs} eq {0};

return "";
end function;


// We now give code that can prove that the trilinear form we gave is the correct one. Choose two random elements that
// generate the subgroup 2^6.W(E6), and then build the trilinear form relations. We also check that l1 and l2 lie in
// NGT.

function CheckE6Form()

GG:=GroupOfLieType("E6",q);
W:=VectorSpace(F,6);
rho:=StandardRepresentation(GG);
Over:=GL(27,q);
g1:=elt<GG|W![z,1,1,1,1,1]>;
g2:=elt<GG|W![1,z,1,1,1,1]>;
g3:=elt<GG|W![1,1,z,1,1,1]>;
g4:=elt<GG|W![1,1,1,z,1,1]>;
g5:=elt<GG|W![1,1,1,1,z,1]>;
g6:=elt<GG|W![1,1,1,1,1,z]>;
Refs:=Reflections(GG);
Mats:=[rho(i):i in [g1,g2,g3,g4,g5,g6]];
MatsE:=[rho(i):i in Refs];
NGT:=sub<Over|Mats cat MatsE>;
T:=sub<NGT|Mats>;
E:=sub<NGT|MatsE>;

l1pre in NGT and l2pre in NGT;

repeat y1:=Random(E); y2:=Random(E); until E eq sub<E|y1,y2>;

mat:=[];
for hh in [y1,y2] do
  h:=hh^(x1^-1);
  for nn in [1..#seqs] do aa:=seqs[nn,1]; bb:=seqs[nn,2]; cc:=seqs[nn,3];
    val:=[F!0:i in [1..#seqs]];
    for i in [1..NumberOfRows(l1)] do if(h[aa,i] ne 0) then for j in [1..NumberOfRows(l1)] do if(h[bb,j] ne 0) then for k in [1..NumberOfRows(l1)] do if(h[cc,k] ne 0) then
      val[Position(seqs,Sort([i,j,k]))]+:=h[aa,i]*h[bb,j]*h[cc,k];
    end if; end for; end if; end for; end if; end for;
    val[nn]-:=1; Append(~mat,val); delete val;
  end for;
end for;

ftest:=Nullspace(Transpose(Matrix(F,mat))).1;
return &and[ftest[i] eq f27[i]:i in [1..#seqs]];
end function;

function CheckL()
// First we set up GG as in CheckE6Form, but also the adjoint representation, as we will need that.

GG:=GroupOfLieType("E6",q);
W:=VectorSpace(F,6);
rho:=StandardRepresentation(GG);
Over:=GL(27,q);
g1:=elt<GG|W![z,1,1,1,1,1]>;
g2:=elt<GG|W![1,z,1,1,1,1]>;
g3:=elt<GG|W![1,1,z,1,1,1]>;
g4:=elt<GG|W![1,1,1,z,1,1]>;
g5:=elt<GG|W![1,1,1,1,z,1]>;
g6:=elt<GG|W![1,1,1,1,1,z]>;
Refs:=Reflections(GG);
Mats:=[rho(i):i in [g1,g2,g3,g4,g5,g6]];
MatsE:=[rho(i):i in Refs];
NGT:=sub<Over|Mats cat MatsE>;
T:=sub<NGT|Mats>;
E:=sub<NGT|MatsE>;

rho2:=AdjointRepresentation(GG);
Mats2:=[rho2(i):i in [g1,g2,g3,g4,g5,g6]];
MatsE2:=[rho2(i):i in Refs];
MNGT:=GModule(NGT);
LNGT:=GModule(NGT,Mats2 cat MatsE2);

L:=sub<NGT|l1pre,l2pre>;
L0:=sub<NGT|l1pre^2,l2pre>;
// Any element of order 3 has to come from the Weyl group, so first we find all classes of order 3 in E.
CCE:=ConjugacyClasses(E);
Clas3:=[i[3]:i in CCE|i[1] eq 3];

// Now we check that the three classes have different actions on L(E6).
#{SocleFactors(Restriction(LNGT,sub<E|i>)):i in Clas3} eq 3;

// The one we want has socle of dimension 27 on LNGT.
for i in Clas3 do if(Dimension(Fix(Restriction(LNGT,sub<NGT|i>))) eq 27) then z1:=i; end if; end for;

// Now we let it act on the 13-part of the torus, and look for subgroups of order 39, then those with a regular 13, of trace 1.
Tz1:=sub<NGT|T,z1>;
ZT:=[i`subgroup:i in Subgroups(Tz1:OrderEqual:=39)|not(IsCyclic(i`subgroup))];
ZT2:=[i:i in ZT|BrauerCharacter(GModule(i))[4] eq 1];
&and[IsConjugate(NGT,i,ZT2[1]):i in ZT2];
// So all 13.3s are conjugate in NGT. Since this is contained in a 13.12, anything other than this must
// centralize the 13, hence lie in T. It must also centralize the 3. We now check that there is no
// involution in T centralized by L0. Since the involutions in the torus are in E (as that is 2^6.W(E6))
// we just check that C_E(L0)=1.

Order(Centralizer(E,L0)) eq 1;

return 0;
end function;

"The function CheckE6Form() checks that the trilinear form f27 from E6 is correct";
"The function CheckDetermination() checks that the determination of the possible conjugates into E6 is correct, and that they are contained in copies of G2(3).";
"The function CheckL() checks the statements about L and L0 made in the paper";