/* Here we consider H=PSL(2,27)<F_4(13). Let L=3^3.13 be the Borel subgroup of H. The best way to proceed here
   seemed to be to construct two copies of H, H1 and H2, that intersect in a fixed copy of L, and together
   generate a copy of F4(8). Once we have this, we are able to prove that there is a unique H1 containing L
   among C_{GL(26,k)}(L)-conjugates and contained in G. Since |N_G(L):N_H(L)|=3, this yields exactly three
   subgroups H1 such that L<H1<G.
*/

// Start by setting up L, H1 and H2.

F<ww>:=GF(64);

l1:=GL(26,F)![
[ww^57,1,ww^59,ww^60,ww^28,ww^30,ww^23,ww^41,ww^16,ww^51,ww^40,ww^7,ww^38,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^57,ww^54,ww^40,ww^41,ww^11,ww^62,ww^22,ww^45,ww^13,ww^47,1,ww^37,ww^13,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^18,ww^3,ww^26,ww^13,ww^52,ww^12,ww^33,ww^10,0,ww^17,ww^5,ww^24,ww^21,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww,ww^52,ww^32,ww^22,ww^47,ww^28,ww^8,ww^55,ww^34,ww^19,ww^49,ww^3,ww^21,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^13,ww^40,ww^42,ww^58,ww^60,ww^62,ww^4,ww^51,ww^25,ww^40,ww^16,ww^40,ww^2,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^61,ww^14,ww^10,ww^45,ww^49,ww^29,ww^54,ww^62,ww^51,ww^26,ww^8,ww^23,ww^38,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^30,ww^43,ww^31,ww^60,ww^34,ww^45,ww^51,ww^48,1,ww^47,ww^32,ww^26,ww^41,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^54,ww^47,ww^54,ww^34,ww^31,ww^19,ww^49,ww^17,ww^14,ww^5,ww^30,ww^29,ww^49,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^49,ww^36,ww^24,ww^46,ww^6,ww^8,ww^50,ww^56,ww^29,ww^59,ww^50,ww^17,ww^34,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^17,ww^19,ww^38,ww^51,ww^11,ww^30,ww^29,ww^18,ww^57,ww^52,ww^51,ww^27,ww^22,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^53,ww^50,ww^31,ww^2,ww^47,ww^49,ww^61,ww^55,ww^60,ww^54,ww^31,ww^21,ww^53,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^16,ww^56,ww^12,ww^26,ww^31,ww^16,ww^51,ww^35,ww^3,ww^34,ww^27,ww^30,ww^29,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^16,ww^30,ww^42,ww^45,ww^9,0,ww^41,ww,ww^61,ww^31,ww^22,ww^35,ww^53,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^58,ww^15,ww^35,ww^10,ww^45,ww^13,ww^11,ww^52,ww^22,ww^57,ww^33,ww^39,ww^34],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^44,ww^9,ww^2,ww^49,ww^5,ww^16,ww^44,ww^48,ww^27,ww,ww^17,ww^20,ww^2],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^47,ww^51,ww^32,ww^56,ww^61,ww^58,ww^28,ww^56,ww^52,ww^43,ww^10,ww^17,ww^13],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^31,ww^60,ww^45,ww^31,ww^38,ww,ww^25,ww^41,ww^23,ww^22,ww^20,ww^61,ww^40],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^19,ww^5,ww^55,ww^42,ww^27,ww^8,ww^23,ww^22,ww^9,ww^19,ww^47,ww^32,ww^7],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^43,ww^50,ww^29,ww^57,ww^37,ww^11,ww^29,ww^55,ww,ww^35,ww^35,ww,ww^33],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^28,ww^45,ww^34,ww^46,ww^24,ww^8,ww^35,ww^58,ww^17,ww^41,ww^28,ww^54,ww^22],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^36,ww^6,ww^6,ww^21,ww^39,ww^7,ww^10,ww^36,ww^9,ww^4,ww^62,ww^61,ww^14],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^6,ww^12,ww^46,ww^28,ww^41,ww^56,ww^44,ww^28,ww^37,ww^45,ww^55,ww^59,ww^20],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^53,ww^36,ww^54,ww^12,ww^56,ww^11,ww^28,ww^8,ww^4,ww^44,ww^17,ww^44,ww^29],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^22,ww^42,ww^16,ww^55,ww^19,ww^43,ww^14,ww^56,ww^47,ww^47,ww^56,ww^30,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^57,ww^38,ww^29,ww^38,1,ww^8,ww^29,ww^40,ww^53,ww^53,ww^24,ww^54,ww^8],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^19,ww^20,ww^31,ww^5,ww^9,ww^38,ww^33,ww^14,ww^2,ww^9,ww^55,ww^58,ww^51]];

l2:=GL(26,F)![
[ww^44,ww^40,ww^7,ww^59,ww^27,ww^51,ww^61,ww^55,ww^28,ww^34,ww^9,ww^4,ww^53,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^12,ww^4,ww^8,1,ww^44,ww^57,ww^62,ww^45,ww^12,ww^49,ww^41,ww^56,ww^43,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^8,ww^32,ww^35,ww^60,ww^50,ww^27,ww^3,ww^50,ww,ww^2,ww^25,ww^16,ww^55,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^33,ww^52,ww^11,ww^21,ww^35,ww^52,ww^13,ww^32,ww^12,ww^57,ww^3,0,ww^62,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^7,ww^52,ww^23,ww^4,ww^35,ww^8,ww^58,ww,ww^30,ww^29,ww^60,ww^30,ww^58,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^18,ww^29,ww^47,ww^33,ww^46,ww^37,ww^21,ww^14,ww^50,ww^59,ww^36,ww^28,ww^29,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^41,ww^61,ww^14,ww,ww^53,ww^44,ww^45,ww^19,ww^29,ww^33,ww^48,ww^54,ww^50,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^55,ww^55,ww^36,ww^58,ww^22,ww^4,ww^49,ww^6,ww^53,ww^18,ww^4,ww^37,ww^15,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^37,ww^47,ww^14,ww^32,ww^20,ww^24,ww^3,ww^11,ww^2,ww^52,ww^23,ww^38,ww^8,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^58,ww^59,ww^59,ww^23,ww^11,ww^18,ww^16,ww^24,ww^61,ww^56,ww^16,ww^40,ww^18,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^33,ww^44,ww^2,ww^26,ww^60,ww^59,ww^30,ww^3,ww^16,ww^49,ww^58,ww^50,ww^6,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^35,ww^46,ww^21,ww^47,ww^50,ww^40,ww^58,ww^62,ww^30,ww^23,ww^13,ww^35,ww^49,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^24,ww^58,ww^31,ww^32,ww^31,ww^58,ww^31,ww^42,ww,ww^29,ww^27,ww^52,ww^37,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^56,ww^23,ww^55,ww^16,ww^25,0,ww^13,ww^4,ww^28,ww^27,ww^25,ww^16,ww^25],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^22,ww^14,ww^40,ww^13,ww^62,ww^62,ww^57,ww^38,ww^62,ww^55,ww^43,ww^27,ww^49],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^37,ww^54,ww^54,ww^56,0,ww^62,ww^55,ww^52,ww^48,ww^57,ww^58,ww^10,ww^33],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^52,ww^32,ww^54,ww^3,ww^51,ww^2,ww^7,ww^39,ww^61,ww^5,ww^40,ww^26,ww^55],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^9,1,ww^56,ww^25,ww^48,ww^17,ww^17,ww^20,ww^35,ww^8,ww^23,ww^25,ww^19],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^28,0,ww^52,ww^35,ww^28,ww^19,ww^17,ww^18,ww^43,ww^42,ww^2,ww^49,ww^29],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^20,ww^53,ww^24,ww^49,ww^17,ww^53,ww^48,ww^49,ww^44,ww^19,ww^54,ww^24,ww],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^23,ww^33,ww^25,ww^38,ww^61,ww^48,1,ww^30,ww^15,ww^57,ww^7,ww^58,ww^17],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^35,ww^4,1,ww^35,ww^59,ww^14,ww^12,ww^7,ww^33,ww^6,ww^35,1,ww^12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^45,ww^36,ww^22,ww^23,ww^51,ww^44,ww^37,ww^35,ww^48,ww^60,ww^38,ww^5,ww^38],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^33,ww^37,ww^58,ww,ww^39,ww^29,ww^25,ww^32,ww^13,ww^33,ww^50,ww^16,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^5,ww^59,ww^62,ww^3,ww^35,ww^9,1,ww^5,ww^56,ww^22,ww^18,ww^50,ww^28],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww,ww^57,ww^52,ww^25,ww^38,ww^50,ww^4,ww^48,ww^53,ww^48,ww^42,ww^19,ww^42]];

l3:=GL(26,F)![[ww^52,1,ww^44,ww^12,ww^13,ww^30,ww^24,ww^11,ww^21,ww^2,ww^18,ww^58,ww^30,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^20,ww^12,ww^8,ww^3,0,ww^12,ww^50,ww^47,ww^4,ww^13,ww^39,0,ww^25,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww,ww^8,ww^49,ww^50,ww^18,ww^23,ww^6,ww^47,ww^23,ww^62,ww^58,ww^48,ww^24,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww,ww^7,ww^13,ww^61,ww^24,ww^13,ww^3,ww^36,ww^36,ww^5,ww^51,ww^48,ww^45,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^46,ww^3,ww^60,ww^2,ww^61,ww^8,ww^58,ww^28,ww^51,ww^38,ww^27,ww^27,ww^31,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^5,ww^12,ww^16,ww^59,ww^51,ww^29,ww^31,ww^42,ww^5,ww^13,ww^38,ww^61,ww^52,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^20,ww^62,ww^18,ww^25,ww^26,ww^37,ww^42,ww^18,ww^5,ww^8,ww^24,ww^58,ww^46,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^32,ww^47,ww^26,ww^39,ww^22,ww^20,ww^46,ww^35,ww^2,ww^27,ww^30,ww^7,ww^44,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^21,0,ww^43,ww^22,ww^61,ww^27,ww^19,ww^15,0,ww^17,ww^23,0,ww^7,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^31,ww^56,ww^15,ww^20,ww^46,ww,ww^23,ww^21,ww^41,ww^53,ww^12,ww^32,ww^12,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^58,ww^24,ww^40,ww^43,ww^60,ww^29,ww^51,ww^40,ww^5,ww^28,ww^4,ww^15,ww^5,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^33,ww^39,ww^20,ww^11,ww^26,ww^12,ww^57,ww^61,ww^34,ww^34,ww^23,1,ww^24,0,0,0,0,0,0,0,0,0,0,0,0,0],
[ww^30,ww^16,ww^7,ww^30,ww^50,ww^40,ww^59,ww^47,ww^48,ww^4,ww^47,ww^23,ww^28,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^40,ww^24,ww^27,ww^46,ww^45,ww^41,ww^23,ww^21,ww^57,ww^22,ww^5,ww^57,ww^6],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,ww^4,ww^31,ww^40,ww^18,ww^44,ww^34,ww^16,ww^55,ww^57,ww^3,ww^4,ww^25],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^41,ww^38,ww^22,ww^10,ww^9,ww^40,ww^16,ww^46,ww^17,ww^58,ww^27,ww^18,ww^56],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^39,ww^24,ww^56,ww^22,ww^58,ww^59,ww^34,ww^18,ww^59,ww^36,ww,ww^12,ww^32],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^3,ww^27,ww^20,ww^4,ww^60,ww^8,ww^42,ww^56,0,ww^55,ww^57,ww^33,ww^9],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^20,ww^38,0,ww^6,ww^41,ww^53,ww^3,ww^47,ww^53,ww^28,ww^34,ww^55,ww^57],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^29,ww^14,ww^37,ww^56,ww^27,ww^43,ww^5,ww^61,ww^55,ww^39,ww^8,ww^38,ww^2],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^33,ww^34,ww^50,ww^62,ww^52,ww^27,ww^2,ww^51,ww^44,ww^5,ww^52,ww^33,ww^40],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^27,ww^50,ww^19,ww^52,ww^29,ww^54,ww^39,ww^22,ww^20,0,ww^37,ww^57,ww^62],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^47,ww^37,ww^21,ww^8,ww,ww^36,ww^52,ww^15,ww^32,ww^33,ww^59,ww^7,ww^49],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^13,ww^51,ww^3,ww^50,ww^41,ww^6,ww^31,ww^24,ww^3,ww^18,ww^5,ww^9,ww],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^57,ww^37,ww^20,ww^53,ww^38,ww^42,ww^32,ww^61,ww^38,ww^49,ww^61,ww^56,ww^58],
[0,0,0,0,0,0,0,0,0,0,0,0,0,ww^35,ww^25,ww^34,ww^54,ww^33,ww^59,ww^41,ww^38,ww^50,ww^10,ww^8,ww^35,ww^43]];

h1:=GL(26,F)![
[ww^55,ww^34,ww^56,ww^37,ww^24,ww^42,ww^62,ww^9,ww^35,ww^60,ww^47,ww^10,ww^49,ww^2,ww^5,ww^44,ww^31,ww^29,ww^3,ww^27,ww^20,ww^7,ww^55,ww^44,ww^54,ww^8],
[ww^23,1,ww^60,ww^19,ww^21,ww^6,ww^23,ww^27,ww^4,ww^39,0,ww^10,ww^23,ww^39,ww^11,1,ww^10,ww^57,ww^12,ww^59,ww^15,ww^53,ww^20,ww^41,ww^9,ww^41],
[ww^43,ww^17,ww^28,0,ww^55,ww^9,ww^40,ww^25,ww^29,ww^11,ww^25,ww^49,ww^14,ww^5,ww^57,ww^56,ww^56,ww^28,ww^31,ww^40,ww^50,ww^37,ww^25,ww^19,ww^41,ww^18],
[ww^12,ww^61,ww^43,ww^59,ww^23,ww^13,ww^61,ww^6,ww^10,ww^29,ww^42,ww^44,ww^26,ww^23,ww^16,ww^58,ww^15,ww^19,ww^3,ww^47,ww^55,ww^46,ww^18,ww^14,ww^40,ww^56],
[ww^60,ww^25,ww^2,ww^30,ww^4,ww^53,ww^33,ww^38,ww^38,ww^25,ww^2,ww^31,ww^43,ww^40,ww^20,ww^16,ww^56,ww^24,ww^61,ww^28,ww^62,ww^53,ww^41,ww^56,ww^36,ww^52],
[ww^34,ww^27,ww^40,ww^9,ww^36,ww^17,ww,ww^45,ww^3,ww^3,ww^36,ww^19,ww^40,ww^58,ww^5,ww^34,ww^18,ww^37,ww^33,ww^15,ww^14,ww^24,ww^35,ww^56,ww^16,ww^10],
[ww^7,1,ww^22,ww^5,ww^3,ww^29,ww^39,ww^35,ww^41,ww^47,ww^17,ww^41,ww^33,ww^35,ww^38,ww^14,ww^16,ww^36,ww^29,ww^14,ww^36,ww^42,ww^10,ww^48,ww^15,ww^59],
[ww^61,ww^62,ww^61,ww^22,ww^38,ww^33,ww^59,ww^55,ww^40,ww^34,ww^42,ww^28,ww^20,ww^29,ww^49,ww^13,ww^38,ww^13,ww^51,ww^55,ww^52,ww^14,ww^36,ww^34,ww,ww^60],
[ww^3,ww^55,ww^36,ww^40,0,ww^17,ww^23,ww^42,ww^43,ww^16,ww^53,ww^32,ww^3,ww^34,ww^26,ww^6,ww^39,ww^30,ww^18,ww^21,ww^31,ww^24,ww^48,ww^12,ww^33,ww^53],
[ww^34,ww,ww^40,ww^34,ww^4,ww^60,ww^61,ww^3,ww^60,ww^28,ww^33,ww^60,ww^34,ww^43,ww^35,ww^32,1,ww^41,ww^35,ww^58,ww^23,ww^14,ww^5,ww^38,ww^14,ww^52],
[ww^28,ww^33,ww^3,ww^61,ww^7,ww^52,ww^21,0,ww^42,ww^5,ww^43,ww^21,ww^23,ww^13,ww^39,ww^42,ww^25,ww^48,ww^4,ww^53,ww^41,ww^13,ww^15,ww^55,ww^45,ww^42],
[ww^51,ww^23,ww^25,0,ww^30,ww^10,ww^27,ww^30,ww^43,ww^50,ww^40,ww^59,ww^25,ww^39,ww^11,ww^37,ww^34,ww^40,ww^35,ww^26,ww^10,ww^24,ww,ww^53,ww^29,ww^41],
[ww^59,ww^55,ww^19,ww^4,ww^56,ww^60,ww^62,ww^4,ww^55,ww^62,ww^27,ww^42,ww^43,ww^49,ww^59,ww^51,ww^35,ww,ww,ww^37,ww^52,ww^51,ww^42,ww^10,ww^24,0],
[ww^52,0,ww^48,ww^51,ww^39,ww^56,ww^25,ww^6,ww^24,ww^37,ww^36,ww^26,ww^10,ww^58,ww^35,ww,ww^17,ww^60,ww^58,ww^44,ww^23,ww^11,ww^58,ww^20,ww^58,ww^5],
[ww^23,ww^25,ww^38,ww^12,ww^39,ww^53,ww^18,ww^53,ww^48,ww^55,ww^56,ww^43,ww^8,ww^35,ww^53,ww^2,ww^46,ww^31,ww^21,ww^14,ww^46,ww^20,ww^57,ww^18,ww^44,ww^16],
[ww^48,ww^14,1,ww^42,ww^21,ww^52,ww^19,ww^25,ww^59,ww^57,ww^52,ww^32,1,ww^2,ww^62,ww^32,ww^38,ww^25,ww^43,ww^18,ww^46,0,ww^40,ww^8,ww^23,ww^52],
[ww^43,ww^54,ww^59,ww^26,ww^6,ww^27,ww^14,ww^48,ww^27,ww^15,ww^32,ww^18,ww^34,ww^3,ww^7,ww^60,ww^16,ww^17,ww^55,ww^61,ww^18,ww^43,ww^46,ww^33,ww^21,ww^17],
[ww^13,ww^16,ww^36,ww^25,ww^31,ww^39,ww^28,1,ww^51,ww^16,ww^51,ww^39,ww^61,ww^33,ww^16,ww^59,ww^23,ww^29,ww^14,ww^4,ww^2,ww^54,ww^22,ww^4,ww^32,ww^62],
[ww^61,ww^2,ww^18,ww^19,ww^2,ww^44,ww^18,ww^50,ww^30,ww^53,ww^53,ww^23,ww^28,ww^35,ww^42,ww^10,ww^41,ww^23,ww^60,ww^31,ww^19,ww^58,ww^59,ww^19,ww^47,ww^17],
[ww^47,ww^58,ww^36,ww^51,ww^19,ww^22,ww^46,ww^26,ww^46,ww^30,ww^17,ww^7,ww^60,ww^50,ww^31,ww^7,ww^59,ww^7,ww^33,ww^45,ww^59,ww^20,ww^9,ww^13,ww^42,ww^17],
[ww^60,ww^58,ww^58,ww^28,ww^35,ww^54,ww^22,ww^39,ww^45,ww^42,ww^59,ww^48,ww^57,ww^21,ww^10,ww^21,ww^48,ww^36,ww^14,ww^34,ww^55,ww^53,ww^62,ww^13,ww^36,ww^23],
[ww^45,ww^44,ww^13,0,ww^4,ww^40,ww^33,ww^24,ww^52,ww^4,ww^60,0,ww^49,ww^3,ww^58,ww^15,ww^35,ww^43,ww^54,ww^5,ww^22,ww^28,ww^42,ww^43,ww^33,ww^59],
[ww^20,ww^61,ww^47,ww^11,ww^52,ww^17,ww^12,ww^54,ww^2,ww^50,ww^34,ww^24,ww,ww^38,ww^56,ww^37,ww^4,ww^23,ww^37,ww^56,ww^16,ww^35,ww^35,ww^23,ww^62,ww^62],
[ww^4,1,ww^32,ww^14,ww^26,ww^58,ww^49,ww^5,ww^28,ww,ww^24,ww^37,ww^10,ww^40,ww^4,ww^16,ww^59,ww^50,ww^2,ww^49,1,ww^2,ww^26,ww^50,ww^24,ww^3],
[ww^16,ww^51,ww^41,1,ww^54,ww^3,ww^2,ww^60,ww^58,ww^26,ww^56,ww^45,ww^47,ww^12,ww^35,ww^22,ww^36,ww^53,ww^31,ww^51,ww^4,ww^10,ww^59,ww^23,ww^20,ww^57],
[ww^59,ww^33,ww^16,ww^15,ww^3,ww^13,ww^18,ww^42,ww^8,ww^33,ww^10,ww^13,ww^28,ww^29,ww^43,ww^4,ww^20,ww^53,ww^10,ww^7,0,ww^48,ww^30,ww^53,ww^59,ww^2]];

h2:=GL(26,F)![
[ww^57,ww^48,ww^47,ww^13,ww^21,ww^57,ww^3,ww^41,ww^55,ww^13,ww^62,ww^59,ww^53,ww^16,ww^60,ww^36,ww^19,ww^50,ww^49,ww^22,ww^29,1,ww^57,ww^38,ww^61,ww^17],
[ww^50,ww,ww^49,ww^37,ww^31,ww^18,ww^46,ww^10,ww^10,ww^12,ww^54,ww^2,ww^45,ww^12,ww^10,ww^14,ww^53,ww^3,ww^15,ww^22,ww^41,ww^15,ww,ww^54,ww^55,ww^20],
[1,ww^60,ww^14,ww^58,ww^44,ww^9,ww^6,ww^14,ww^18,ww^5,ww^6,ww^25,ww^4,ww^52,ww^62,ww^33,ww^4,ww^8,ww^42,ww^39,ww^7,ww^26,ww^22,ww^26,ww^53,ww^15],
[ww^38,ww^31,ww^43,ww^61,ww^61,ww^36,ww^32,ww^8,ww^3,1,ww^17,ww^33,ww^40,ww^5,ww^16,ww^54,ww^30,ww^37,ww^17,ww^52,ww^12,ww^58,ww^34,ww^51,ww^31,ww^29],
[ww^13,ww^54,ww^16,ww^6,ww^18,1,ww^17,ww^8,1,ww^61,ww^41,ww^25,ww^9,ww^22,ww^25,ww^39,ww^9,ww^24,ww^38,ww^50,ww^34,ww^19,ww^7,ww^8,ww^58,ww^15],
[ww^20,ww^3,ww^35,ww^14,ww^22,ww^42,ww^2,ww^37,ww^11,ww^48,ww^54,ww^22,ww^47,ww^54,ww^30,ww^6,ww^34,0,ww^40,ww^3,ww^50,ww^8,ww^16,ww^7,ww^52,ww^5],
[ww^24,ww^36,ww^13,ww^32,ww^22,ww^54,ww^55,ww^19,ww^48,ww^23,ww^51,ww^10,ww^15,ww^37,ww^10,ww^12,ww^49,ww^61,ww^2,ww^60,ww^42,ww^42,ww^13,ww^62,ww^27,ww^35],
[ww^33,ww^35,ww^33,ww^36,ww^10,ww^37,ww^29,ww^37,ww^47,ww^48,ww^28,ww^22,ww^42,ww^50,ww^28,ww^54,ww^29,ww^43,ww^50,ww^11,ww^21,ww^18,ww^40,ww^5,ww^49,ww^32],
[ww^14,ww^31,ww^49,ww^18,ww^40,ww^24,ww^31,ww^3,ww^12,ww,ww^55,ww^51,ww^20,ww^13,ww^25,ww^3,ww^10,ww^10,ww^23,ww^46,ww^39,ww^17,ww^37,ww^45,ww^37,ww^11],
[ww^42,ww^57,ww^57,ww^44,ww^5,ww^42,ww^8,ww^17,1,ww^25,ww^23,ww^22,ww^51,ww^36,ww^12,ww^56,ww^45,ww^3,ww^62,ww^2,ww^47,ww^32,ww^29,ww^19,ww^38,ww^52],
[ww^53,ww^6,ww^35,ww^8,ww^57,ww^54,ww^22,ww^23,ww^34,ww^21,ww^27,ww^24,ww^52,ww^25,ww^44,ww^52,ww^26,ww^5,ww^4,ww^38,ww^35,ww^9,ww^33,ww^17,ww^59,ww^6],
[ww^33,ww^56,ww^45,ww^6,0,ww^46,ww^48,ww^46,ww^28,ww^42,ww^9,ww^17,ww^57,ww^45,ww^50,ww^12,ww,ww^59,1,ww^4,ww^12,ww^36,ww^50,ww^45,ww^37,ww^2],
[ww^23,ww^17,ww^36,ww^9,ww^40,ww^30,ww^24,ww^60,ww^12,ww^46,ww^55,ww^59,ww^38,ww^56,ww^43,ww^25,ww^11,ww^49,ww^12,ww^27,ww^12,ww^13,ww^15,ww^14,1,ww^57],
[ww^9,ww^34,ww^39,ww^4,ww^42,ww^32,ww^60,ww^3,ww^5,ww^32,ww^22,ww^51,ww^40,ww^7,ww^7,ww^35,ww^22,ww^62,ww^60,ww^8,ww^18,ww^16,ww^20,ww^17,0,ww^7],
[ww^58,ww^49,ww^57,ww^20,ww^55,ww^13,ww^59,ww^3,ww^14,ww^46,ww^57,ww^21,ww^32,ww^20,ww^48,ww^53,ww^58,ww^27,ww^22,ww^12,ww^54,ww^45,ww^17,ww^51,ww^59,ww^42],
[ww^38,ww^10,ww^36,ww^10,ww^62,1,ww^4,ww^55,ww^52,ww^36,ww^18,ww^61,ww^13,ww^53,ww^55,ww^14,ww^31,ww^11,ww^58,ww^5,ww^41,ww^3,ww^21,ww^6,ww^51,ww^50],
[ww^57,ww^14,ww^60,ww^2,ww^23,ww^40,ww^34,ww^18,ww^10,ww^37,ww^2,ww^9,ww^31,ww^43,ww^13,1,ww^53,1,ww^19,ww^8,ww^19,ww^56,ww^13,ww^31,ww^19,ww^47],
[ww^49,ww^3,ww^7,ww^22,ww^9,ww^32,ww^11,ww^41,ww^22,ww^16,ww^55,ww^25,ww^7,ww^54,ww^16,ww^49,ww^35,ww^3,ww^59,ww^41,ww^35,ww^53,ww^18,ww^57,ww^27,ww^17],
[ww^34,ww^57,ww^20,ww^47,ww^48,ww^12,ww^51,ww^45,ww,ww^54,ww^11,ww^2,ww^43,ww^23,1,ww^7,ww^55,ww,ww^49,ww^42,ww^6,ww^14,ww^4,ww^62,ww^17,ww^33],
[ww^57,ww^17,ww^27,ww^11,ww,ww^25,ww^34,ww^36,ww^17,ww^30,ww^3,ww^51,ww^46,ww^38,ww^48,ww^31,ww^38,ww^53,ww^32,ww^46,ww^54,ww^59,ww^24,ww^5,ww^59,ww^43],
[ww^20,ww^16,ww^8,ww^42,ww^59,ww^53,ww^62,ww^16,ww^42,ww^44,ww^44,ww^14,ww^40,ww^60,ww^43,0,ww^61,ww^30,ww^17,ww^29,ww^11,ww^12,ww^20,ww^20,ww^49,ww^3],
[ww^12,ww^49,ww^53,ww^49,ww^59,ww^5,ww^15,ww^53,ww^31,ww^7,ww^22,ww^54,ww^27,ww^28,ww^33,ww^49,ww^45,ww^9,ww^34,ww^14,ww^44,ww^54,0,ww^33,ww^19,ww^17],
[ww^44,ww^8,ww^32,ww^22,ww^4,ww^26,ww^2,ww^35,ww^42,ww^19,ww^3,ww^10,ww^40,ww^14,ww^53,ww^33,ww^17,ww^25,ww^49,1,ww^3,ww^47,ww^18,ww^48,ww^58,ww^5],
[ww^32,ww^8,ww^6,ww^6,ww^56,ww^8,ww^56,ww^6,ww^37,0,ww^32,ww^18,ww^5,ww^48,ww^10,ww^30,ww^5,ww^42,ww^25,ww^34,ww^8,ww^34,0,ww^2,ww^7,ww^55],
[ww^33,ww^21,ww^30,ww^57,ww^2,ww^37,ww^25,ww^8,ww^60,ww,ww^28,ww^31,ww^46,ww^43,ww^30,ww^60,ww^58,ww^20,ww^43,ww^23,ww^59,ww^16,ww^26,ww^40,ww^43,ww^29],
[ww^59,ww^57,ww^2,ww^22,ww^28,1,ww^49,ww^8,ww^44,ww^33,ww^23,ww^12,ww^23,ww^16,ww^13,ww^23,ww^61,ww^12,ww^34,ww^10,ww^10,ww^36,ww^9,ww^8,ww^26,ww^29]];

G:=sub<GL(26,F)|l1,l2,h1,h2>;
H1:=sub<GL(26,F)|l1,l2,h1>;
H2:=sub<GL(26,F)|l1,l2,h2>;
L:=sub<GL(26,F)|l1,l2>;
NL:=sub<GL(26,F)|l1,l2,l3>;

NG:=sub<GL(26,F)|l1,l2,h1,h2,l3>;

// We need the indices of the trilinear form as well.
seqs:=[[i,j,k]:i,j,k in [1..26]|i le j and j le k];

/* We next give the symmetric trilinear form that comes from F4. This was found by choosing elements of F4, and then
using the G-invariance of the form to obtain constraints on the structure constants of it. Code is included at the end
to reconstruct this, but it can take a while, so we give it here for ease of calculations.
*/

f26:=
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,ww^24,ww^11,ww^34,ww^48,ww^33,ww^55,ww,ww^14,ww^12,ww^3,ww^4,ww^27,ww^28,ww^14,ww^11,ww^43,ww^37,ww^54,ww^34,ww^36,ww^24,ww^31,
ww^3,0,ww^34,ww^8,ww^45,ww^5,ww^7,ww^42,ww^49,ww^13,ww^3,ww^56,ww^36,ww^51,ww^47,ww^3,ww^2,ww^59,ww^12,ww^30,ww^61,ww^36,ww^33,ww^45,ww^14,0,ww^28,ww^41,ww^35,ww^43,ww^44,ww^22,ww^56,ww^31,ww^56,ww^17,
ww^31,ww^42,ww^43,ww^26,ww^47,ww^3,ww^49,ww^19,ww^39,ww^54,ww^34,ww^37,0,ww^9,1,ww^46,ww^44,ww^11,ww^62,ww^14,ww^36,ww^60,ww^40,ww^25,ww^43,ww^21,ww^18,ww^12,ww^25,ww^11,ww^16,ww^13,ww^27,ww^60,0,ww^52,
ww,ww^36,ww^27,ww^19,ww^18,ww^33,ww^51,ww^51,ww^8,ww^26,ww^48,ww^8,ww^34,ww^8,ww^17,ww^55,ww^10,ww^18,ww^45,0,ww^2,ww^26,0,ww^3,ww^14,ww^5,ww^14,ww^24,ww^54,ww^32,ww^30,ww^28,ww^10,ww^58,ww^9,ww,0,
ww^51,ww^26,0,ww^23,ww^51,ww^43,ww^53,1,ww^47,ww^44,ww^52,ww^5,ww^23,ww^34,ww^37,ww^53,ww^5,ww^41,ww^26,ww^20,ww^58,0,ww^17,ww^55,ww^2,ww^41,ww^23,ww^31,ww^16,ww^46,ww^19,ww^59,ww^30,ww^27,ww^3,ww^41,
ww^57,ww,ww^15,0,ww^45,ww^10,ww^52,ww^33,ww^31,ww^6,ww^25,ww^37,ww^49,ww^56,ww^19,ww^40,ww^19,ww^47,ww^6,ww,0,0,ww^46,ww^39,ww^33,ww^50,ww^9,ww^8,ww^11,ww^47,ww^56,ww^20,ww^18,ww^23,ww^48,ww^23,0,
ww^58,ww^32,ww^31,ww^4,ww^61,ww^50,0,ww^36,ww^29,ww^49,ww^19,ww^17,ww^7,ww^18,0,ww^10,ww^19,ww^5,ww^57,ww^38,ww^15,ww^43,ww^24,ww^47,ww^11,ww^27,ww^33,ww^57,0,ww^61,ww^2,ww^4,1,1,ww^24,ww^37,ww^9,
ww^41,ww^25,ww^34,ww^4,0,ww^25,ww^42,ww^11,ww^4,ww^43,ww^54,ww^58,ww^34,0,ww^7,ww^53,0,ww^50,ww^23,ww^33,ww^51,ww^56,ww^61,ww^40,ww^51,ww^51,ww^43,0,1,ww^60,ww^41,ww^40,ww^25,ww^31,ww^51,ww^49,ww^57,
0,ww^48,ww^52,ww^48,ww^11,ww^51,ww^62,ww^29,ww^41,0,ww^12,1,ww^27,ww^48,ww^31,ww^6,ww^39,0,ww^51,ww^42,ww^25,ww^23,ww^7,ww^46,0,ww^54,ww^55,ww^34,ww^7,ww^2,0,0,ww^11,ww^11,ww^51,0,ww^42,ww^61,ww^27,
0,ww^21,ww^61,0,ww^49,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,ww^28,ww^28,ww^58,ww^38,ww^17,ww^24,ww^3,0,ww^53,ww^49,ww^43,ww^12,ww^48,ww^18,ww^43,ww^20,ww^33,1,ww^55,
ww^49,ww^17,ww^39,ww^28,0,ww^28,ww^41,ww^56,ww^39,ww^49,ww^12,ww^31,ww^45,ww^57,ww^57,ww^15,ww^16,ww^28,ww^34,ww^6,ww^51,ww,ww^50,ww^55,ww^24,ww^54,ww^28,0,ww^60,ww^52,ww^2,ww^55,ww^38,ww^47,ww^3,ww^62,
ww^35,ww^36,ww^55,ww^43,ww^58,ww^26,ww^23,ww^48,ww^15,ww^45,0,ww^2,ww^40,0,ww^46,ww^16,ww^38,ww^57,ww^6,ww^5,ww^30,ww^19,ww^33,ww^38,ww^38,ww^27,ww^35,ww^21,ww^44,ww^9,ww^24,ww^51,ww^2,ww^40,0,ww^56,
ww^12,ww^52,ww^23,ww^59,ww^11,ww^62,ww^44,ww^11,0,ww^58,1,ww^14,ww^61,ww^47,ww^17,ww^10,ww^55,1,0,ww^22,ww^51,ww^14,ww^44,ww^29,ww^59,ww^56,ww^17,ww^49,ww^18,ww^61,ww^16,ww^17,ww^61,ww^42,ww^15,ww^45,
ww^11,0,ww^25,ww^37,ww^57,ww^3,ww^42,ww^44,1,ww^49,ww^13,ww^18,ww^9,ww^4,ww^33,ww^17,ww^49,ww^12,ww^55,0,ww^56,ww^8,ww^6,ww^52,ww^24,ww^11,ww^52,ww^34,ww^14,ww^41,ww^2,ww^31,ww^25,ww^6,ww^10,ww^45,0,
ww^47,ww^25,ww^17,ww^11,ww^27,ww^58,ww^58,ww,ww^10,ww^26,ww^28,ww^30,ww^5,ww^16,ww^34,0,ww^51,ww^8,ww^51,ww^20,ww^33,ww^55,ww^61,ww^14,ww^24,ww^48,ww^29,ww^13,ww^17,ww^10,0,ww^21,ww^8,ww^28,1,1,ww^60,
ww^26,1,ww^33,ww^60,1,ww^24,ww^36,0,ww^10,ww^27,ww^24,ww^9,ww^42,ww^55,ww^27,ww^50,ww^31,ww^52,ww^38,ww^40,0,ww^19,ww^60,ww^7,ww^43,ww^50,ww^62,ww^31,ww^33,ww^16,ww^37,ww^19,0,ww,ww^6,ww^34,ww^56,
ww^42,ww^20,ww^37,ww^39,ww^15,ww^31,0,ww^32,ww^14,ww^12,ww^51,ww^26,ww^24,ww^10,ww^53,ww^14,0,ww^28,ww,ww^46,ww^51,ww^29,ww^53,ww^32,ww^44,0,ww^4,ww^34,ww^62,ww^19,ww^10,ww^4,ww^11,0,ww^5,ww^43,ww^5,
ww^26,ww^57,ww^13,0,ww^33,ww^2,ww^6,ww^55,ww^55,0,ww^32,1,ww^15,ww,0,ww^7,ww^17,ww^22,0,ww^11,ww^26,0,ww^30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,ww^14,ww^49,ww^6,ww,
ww^14,ww^12,ww^22,1,ww^38,ww^50,ww^7,ww^58,ww^42,ww^50,ww^60,ww^33,ww^25,ww^49,ww^18,ww^30,ww^37,ww^6,0,ww^22,ww^20,ww^16,ww^8,ww^12,ww^25,ww^46,ww^3,ww^26,ww^20,ww^6,ww^25,ww^11,ww^27,ww^21,ww^28,
ww^14,ww^21,ww^37,ww^57,ww^34,0,ww^59,ww^48,ww^59,ww^16,ww^43,0,ww^46,ww^43,ww^19,ww^3,ww^15,ww^11,ww^29,ww^40,ww^20,ww^57,ww^17,ww^10,ww^36,ww^39,0,ww^49,ww^32,ww^59,ww^51,ww^39,ww^53,ww^3,ww^21,ww^48,
ww^61,ww^53,ww^12,ww^2,ww^50,ww^25,ww^54,ww^42,ww^24,ww^19,0,ww^12,ww^59,ww^59,ww^61,ww^9,ww^48,ww^8,ww^18,ww^25,ww^48,ww^11,ww^2,ww^18,ww^11,0,ww^45,ww^28,ww^42,0,ww^43,ww^54,ww^19,ww^39,ww^21,ww^40,
ww^35,ww^43,ww^9,ww^8,ww^46,ww^47,ww^39,ww^40,ww^13,0,ww^42,0,ww^8,ww^19,ww^48,ww^5,ww^39,ww^22,0,ww^59,ww,ww^53,ww^10,1,ww^21,ww^37,ww^20,ww^54,0,ww^50,ww^12,0,ww^39,ww^16,ww^4,ww^44,ww^38,ww^48,
ww^35,ww^51,ww^29,ww^27,ww^5,ww^17,0,ww^3,ww^21,ww^60,ww^20,ww^53,ww^11,ww^53,ww^6,ww^7,ww^24,ww^33,ww^31,ww^31,ww^53,0,ww^40,ww^27,ww^49,ww^11,ww^58,ww^32,ww^30,ww^59,ww^33,ww^9,ww^29,ww^47,ww^56,0,
ww^51,ww^54,ww^23,ww^8,ww^58,1,ww^37,ww^16,ww^35,ww^21,ww^22,ww^61,0,0,ww,ww^12,ww^14,ww^36,ww^11,ww^29,ww^15,ww^60,ww^42,ww^28,0,ww^30,0,ww^22,ww^24,ww^40,ww^10,ww^22,ww^35,ww^3,ww^10,0,ww^59,ww^7,
ww^41,ww^49,ww^52,ww^37,ww^3,ww^54,ww^46,0,ww^28,ww^4,ww^35,ww^38,ww^12,ww^14,ww^49,ww^45,0,ww^14,ww^11,ww^46,ww^37,ww^37,ww,ww^17,0,ww^6,ww^19,ww^27,ww^7,ww^16,ww^57,0,ww^37,ww^11,ww^34,ww^24,ww^47,
0,ww^51,ww^28,ww^38,ww^44,0,ww^12,ww^61,ww^27,0,ww^15,ww^29,0,ww^12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,ww^56,ww^41,ww^44,ww^34,ww^62,ww^52,ww^59,ww^10,ww^44,ww^28,ww^10,
ww^20,ww^52,ww^20,ww^31,ww^31,ww^21,ww^43,ww^39,ww^57,ww^39,0,ww^61,ww^48,ww^8,ww^56,ww^33,ww^15,ww^3,ww^59,ww^60,ww^58,1,ww^7,ww^46,ww,ww^50,ww^55,ww^56,ww^25,ww^43,ww^42,0,ww^54,ww^42,ww^33,ww^46,
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ww^40,0,0,0,0,0,0,0,ww^47,ww^61,ww^40,0,ww^60,ww^45,0,ww^35,0,0,0,0,0,0,ww^32,ww^52,0,ww^58,0,0,0,0,0,ww^15,0,0,0,0,0];

ents:=[i:i in [1..#seqs]|f26[i] ne 0];

// Now set up the centralizer of L in GL(26,F), which is given by all possible matrices scal below.
// We also need this matrix ring so we can multiply elements of F4 by scal, as we have to coerce
// them into this ring first.

R<a,b>:=PolynomialRing(F,2);
mats:=MatrixRing(R,NumberOfRows(l1));
scal:=mats!DiagonalMatrix([a,a,a,a,a,a,a,a,a,a,a,a,a,b,b,b,b,b,b,b,b,b,b,b,b,b]);

// Now we introduce some commands so allow us to proceed with the method in the paper.

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]]*f26[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]]*f26[Position(seqs,Sort([i[1],i[2],i[3]]))]:i in intseqs];

return e1-e2;
end function;

// This function determines all subgroups that lie in G and are C_{GL(26,k)}(L)-conjugate to H1. It then checks
// that H1, H2 and G are what I say they are. And then it checks that the three H1s are not all the same.


function CheckDetermination()

ProdRel([1,13,17])/ww^13 eq b*(a-b)*(a+ww^46*b);
ProdRel([2,5,7])/ww^46 eq (a-b)*(a+ww^21*b)*(a+ww^42*b);
// Only solution is a=b.

IsIsomorphic(PSL(2,27),H1) and IsIsomorphic(PSL(2,27),H2);
Index(NL,L) eq 3;
M:=GModule(G);
Dimension(Fix(SymmetricPower(M,3))) eq 1;
// One cannot easily check that the non-abelian factors of type F4 in even characteristic are what they are supposed to be.
// But it stabilizes a symmetric trilinear form, and acts irreducibly, and one may check that its order is equal to that of
// F4(8).

NM:=GModule(NG);
Dimension(Fix(SymmetricPower(NM,3))) eq 1;

H1^l3 eq H2;

return "";
end function;

function CheckF4Form()

mat:=[];
for h in [l1,l2,h1,h2] do
  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(h)] do if(h[aa,i] ne 0) then for j in [1..NumberOfRows(h)] do if(h[bb,j] ne 0) then for k in [1..NumberOfRows(h)] do
      val[Position(seqs,Sort([i,j,k]))]+:=h[aa,i]*h[bb,j]*h[cc,k];
    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 f26[i]:i in [1..#seqs]];
end function;

"The function CheckF4Form() checks that the trilinear form f26 from F4 is correct";
"The function CheckDetermination() checks that the determination of the possible conjugates into F4 is correct.";