F<a,b,c,d,e,f,g> := FreeGroup(7);
G<s,t,u,v,w,x,y> := quo<F|
a^2,b^2,c^2,d^2,e^2,f^2,(a*b)^3,(a*c)^2,(a*d)^2,(a*e)^2,(a*f)^2,
(b*c)^3,(b*d)^2,(b*e)^2,(b*f)^2,(c*d)^8,(c*e)^2,(c*f)^2,
(d*e)^3,(d*f)^2,(e*f)^3,(a*f) = (c*d)^4,
(c*g*c*g,d),(c,d*g*d*g),a*f*g*g,(b*g*b*c*g*c)^2*b,
(e*g*e*d*g*d)^2*e, (b*c*d*g)^5>;
H := sub<G|s,t,u,v,w,x>;
V := CosetSpace(G, H);
ON := CosetImage(V);

/* element of (projective) order */ 

ElementOfOrder := function (P, RequiredOrder, Limit : Fct := Order) 

   if Type (RequiredOrder) eq RngIntElt then  
      RequiredOrder := {RequiredOrder}; 
   end if; 

   NmrTries := Limit; 
   repeat 
      g, w := Random (P); 
      o := Fct (g); 
      NmrTries -:= 1; 
      rem := exists (x) {x : x in RequiredOrder | (o mod x eq 0)}; 
   until rem or (NmrTries eq 0); 

   if rem then return g^(o div x), w^(o div x); else return false, _; end if; 

end function;

/* sporadic maximal subgroup record format */ 

SporadicRF := recformat <Name : MonStgElt, Parent : MonStgElt,  
          generators : SeqEnum, Order : RngIntElt, Index: RngIntElt>;

/* maximal subgroup list production */ 

ListMaximals := function (G, ParentName, L) 
  X := []; Names := []; Orders := []; Index := []; 
    for i in [1..#L] do 
      if assigned L[i]`Parent and L[i]`Parent eq ParentName and #L[i]`generators gt 0 then  
         F := Parent (L[i]`generators[1]); 
         a := F.1; b := F.2; 
         f := hom <F -> G | [G.1, G.2]>; 
         images := [f (w): w in L[i]`generators]; 
         X[#X + 1] := sub <G | images>; 
         Names[#Names + 1] := L[i]`Name; 
         Orders[#Orders + 1] := L[i]`Order; 
         Index[#Index + 1] := L[i]`Index; 
      end if; 
   end for; 
   return X, Names, Orders, Index; 
end function;

/* presentation on standard generators for ON */ 

PresentationON := function (G : UserGenerators := [], Projective := false) 
   Fct := Projective select ProjectiveOrder else Order; 
   if #UserGenerators eq 0 then  
      UserGenerators := [G.i : i in [1..Ngens (G)]]; 
   end if; 
   if #UserGenerators ne 2 then "Require 2 user generators";  
                           return false; end if; 
   a := UserGenerators[1]; b := UserGenerators[2]; 
   "Warning: presentation not available"; 
   return true; 
end function;

/* black box recognition procedure for O'N */ 

StandardGeneratorsON := function (G : Projective := false, Limit := 100) 

   Fct := Projective select ProjectiveOrder else Order; 
    
   P := RandomProcessWithWords (G);  
   g, w := ElementOfOrder (P, {20,28}, Limit: Fct := Fct); 
   if Type (g) eq BoolElt then return false, _; end if; 

   o := Fct (g); 
   x := g^(o div 2); xw := w^(o div 2); 
   b := g^(o div 4); bw := w^(o div 4); 

   nmr := Limit; 
   repeat 
      nmr -:= 1; 
      c, cw := Random (P); 
      a := x^c; 
   until Fct (a * b) eq 11 or nmr eq 0; 

   if nmr eq 0 then return false, _; end if; 

   aw := xw^cw;  

   if not PresentationON (G: UserGenerators := [a, b], Projective := Projective) then  
      return false, _;  
   end if; 
       
   return [a, b], [aw, bw]; 

end function;

/* generators for maximal subgroups of ON */ 

GeneratorsONMax1 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w4*w4; 
w4:=w5^-1; 
w6:=w4*w2; 
w2:=w6*w5; 
w4:=w3^-1; 
w5:=w4*w1; 
w1:=w5*w3; 

   return [w1,w2]; 

end function; 

GeneratorsONMax2 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w4*w4; 
w6:=w5^-1; 
w3:=w2^-1; 
w4:=w2*w1; 
w1:=w4*w3; 
w4:=w6*w3; 
w2:=w4*w5; 

   return [w1,w2]; 

end function; 

GeneratorsONMax3 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w8:=w6*w5; 
w9:=w3*w8; 
w8:=w9*w9; 
w2:=w8*w8; 
w6:=w5^6; 
w7:=w6^-1; 
w8:=w7*w2; 
w2:=w8*w6; 
w5:=w4^7; 
w6:=w5^-1; 
w7:=w6*w1; 
w1:=w7*w5; 

   return [w1,w2]; 

end function; 

GeneratorsONMax4 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w5^10; 
w7:=w6*w2; 
w8:=w2*w6; 
w9:=w8^-1; 
w6:=w9*w7; 
w1:=w6^14; 
w2:=w3*w4; 

   return [w1,w2]; 

end function; 

GeneratorsONMax5 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w7:=w6*w3; 
w8:=w7*w4; 
w9:=w7*w8; 
w2:=w9*w9; 
w1:=w9*w9; 
w7:=w6*w4; 
w8:=w7^19; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w7:=w4*w6; 
w8:=w7^17; 
w6:=w5*w4; 
w7:=w6*w3; 
w9:=w7^8; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

GeneratorsONMax6 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w7:=w6*w3; 
w8:=w7*w4; 
w9:=w7*w8; 
w2:=w9*w9; 
w1:=w9*w9; 
w7:=w6*w4; 
w8:=w7^19; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w8:=w4*w6; 
w6:=w5*w4; 
w7:=w6*w3; 
w9:=w7^12; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

GeneratorsONMax7 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w4:=w5*w2; 
w5:=w4*w4; 
w2:=w5*w5; 
w4:=w3*w3; 
w5:=w3*w4; 
w4:=w5^-1; 
w3:=w4*w1; 
w1:=w3*w5; 

   return [w1,w2]; 

end function; 

GeneratorsONMax8 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w4*w2; 
w4:=w5*w3; 
w3:=w4*w4; 
w2:=w3*w3; 
w4:=w5*w5; 
w3:=w5*w4; 
w4:=w3^-1; 
w5:=w4*w1; 
w1:=w5*w3; 

   return [w1,w2]; 

end function; 

GeneratorsONMax9 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w8:=w6*w5; 
w9:=w3*w8; 
w8:=w9*w9; 
w2:=w8*w8; 
w5:=w4*w4; 
w4:=w5*w5; 
w5:=w4^-1; 
w6:=w5*w2; 
w2:=w6*w4; 
w4:=w3*w3; 
w3:=w4*w4; 
w4:=w3^-1; 
w5:=w4*w1; 
w1:=w5*w3; 

   return [w1,w2]; 

end function; 

GeneratorsONMax10 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w7:=w6*w3; 
w8:=w7*w4; 
w9:=w7*w8; 
w2:=w9*w9; 
w7:=w6*w4; 
w8:=w7^4; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w7:=w4*w6; 
w8:=w7^19; 
w6:=w5*w4; 
w7:=w6*w3; 
w9:=w7^2; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

GeneratorsONMax11 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w7:=w6*w3; 
w8:=w7*w4; 
w9:=w7*w8; 
w2:=w9*w9; 
w7:=w6*w4; 
w8:=w7^14; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w7:=w4*w6; 
w8:=w7^2; 
w6:=w5*w4; 
w9:=w6*w3; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

GeneratorsONMax12 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w9:=w2*w5; 
w2:=w9*w9; 
w8:=w6*w4; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w7:=w4*w6; 
w8:=w7^10; 
w6:=w5*w4; 
w7:=w6*w3; 
w9:=w7^3; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

GeneratorsONMax13 := function (a,b) 

   w1 := a; w2 := b; 

w3:=w1*w2; 
w4:=w3*w2; 
w5:=w3*w4; 
w6:=w3*w5; 
w9:=w2*w5; 
w2:=w9*w9; 
w7:=w6*w4; 
w8:=w7^13; 
w9:=w8^-1; 
w10:=w9*w1; 
w1:=w10*w8; 
w7:=w4*w6; 
w8:=w7^27; 
w6:=w5*w4; 
w7:=w6*w3; 
w9:=w7^3; 
w7:=w8*w9; 
w6:=w7^-1; 
w8:=w6*w2; 
w2:=w8*w7; 

   return [w1,w2]; 

end function; 

/* list of subgroups of ON */ 

DataON := function () 

   F<a, b> := SLPGroup (2); 
    
   L := [ 
   rec <SporadicRF | Name := "ON", generators := [a, b], Order := 460815505920>, 
    
   rec <SporadicRF | Name := "L3(7):2", Parent := "ON", generators :=  
   GeneratorsONMax1(a,b), Order := 3753792, Index := 122760>, 

   rec <SporadicRF | Name := "L3(7):2b", Parent := "ON", generators :=  
   GeneratorsONMax2(a,b), Order := 3753792, Index := 122760>, 

   rec <SporadicRF | Name := "J1", Parent := "ON", generators :=  
   GeneratorsONMax3(a,b), Order := 175560, Index := 2624832>, 

   rec <SporadicRF | Name := "4.L3(4):2", Parent := "ON", generators :=  
   GeneratorsONMax4(a,b), Order := 161280, Index := 2857239>, 

   rec <SporadicRF | Name := "((3^2:4)xA6).2", Parent := "ON", generators :=  
   GeneratorsONMax5(a,b), Order := 25920, Index := 17778376>, 

   rec <SporadicRF | Name := "3^4:2^(1+4).D10", Parent := "ON", generators :=  
   GeneratorsONMax6(a,b), Order := 25920, Index := 17778376>, 

   rec <SporadicRF | Name := "L2(31)", Parent := "ON", generators :=  
   GeneratorsONMax7(a,b), Order := 14880, Index := 30968784>, 

   rec <SporadicRF | Name := "L2(31)b", Parent := "ON", generators :=  
   GeneratorsONMax8(a,b), Order := 14880, Index := 30968784>, 

   rec <SporadicRF | Name := "4^3.L3(2)", Parent := "ON", generators :=  
   GeneratorsONMax9(a,b), Order := 10752, Index := 42858585>, 

   rec <SporadicRF | Name := "M11", Parent := "ON", generators :=  
   GeneratorsONMax10(a,b), Order := 7920, Index := 58183776>, 

   rec <SporadicRF | Name := "M11b", Parent := "ON", generators :=  
   GeneratorsONMax11(a,b), Order := 7920, Index := 58183776>, 

   rec <SporadicRF | Name := "A7", Parent := "ON", generators :=  
   GeneratorsONMax12(a,b), Order := 2520, Index := 182863296>, 

   rec <SporadicRF | Name := "A7b", Parent := "ON", generators :=  
   GeneratorsONMax13(a,b), Order := 2520, Index := 182863296> 

   ]; 
    
   return L; 
    
end function; 

/* code to find standard generators of O'N and produce listing of maximal subgroups */ 

MaximalsON := function (G) 

   x, y := StandardGeneratorsON(G); 
   if Type(x) eq BoolElt then "Unable to find Standard Generators"; return false; end if; 

   G := sub<G|x[1],x[2]>; 
   return ListMaximals(G, "ON", DataON()); 

end function;

m:=MaximalsON(ON);
m;