#We find all primitive subgroups of $GL(16,3)$ with parameters $e=16$, $a=1$
#In the notations of Theorem 5.1, if $H$ is a primitive solvable subgroup, then it possesses a series

#$$1<U<F<A<H,$$

#where $U$ is cyclic of order $2$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+8}$, and $E$ is chosen so that $N_{GL(8,3)}(F)/F\simeq O^+(8,2)$, moreover
#in this case $H=A$. 

#First we make general variables:


gap> z:=Z(3);;
gap> v:=[z,z^2,0*z,z^2,z,z,z^2,z^2,0*z,z,z,z^2,z,z^2,0*z,z^2];;
gap> result:=[];;



#Now we run the following loop, where gens is the list of generators from generators-GL(16,3)+.txt. Each entry of the list result has three items: the first is the
#number of the group in the list gens, the second is the order of the group, and the third is the order of the stabilizer of v in H. 
#If the order of the stabilizer is 1, then |v^H=|H| and Corollary 2.5 can be applied.


gap> for j in [1..Size(gens)] do
> H:=Subgroup(G,gens[j]);;
> n:=Size(H);
> m:=Size(v^H);
> result[j]:=[j,n,n/m];
> od;
gap> result;
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#Now we choose groups such that |v^H|<>|H| in the list res1

gap> res1:=Filtered(result,x->x[3]<>1);;


#We choose another vector v and repeat the cycle for the remaining groups 

gap> v:=[z,z^2,0*z,z^2,z,z,z^2,z^2,0*z,z,z,z^2,0*z,0*z^2,0*z,0*z^2];;
gap> result1:=[];;
gap> for j in res1 do
> H:=Subgroup(GL(16,3),gens[j[1]]);;
> n:=j[2];;
> m:=Size(v^H);;
> Append(result1,[[j[1],j[2],n/m]]);
> Display([j[1],j[2],n/m]);
> od;
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[ 532, 1327104, 4 ]
[ 533, 1327104, 2 ]
[ 534, 1327104, 2 ]
[ 535, 1327104, 2 ]
[ 536, 1327104, 4 ]
[ 537, 1327104, 2 ]
[ 538, 1327104, 4 ]
[ 540, 1327104, 2 ]
[ 541, 1990656, 4 ]
[ 542, 1990656, 2 ]
[ 544, 1990656, 4 ]
[ 545, 1990656, 2 ]
[ 546, 1990656, 2 ]
[ 547, 1990656, 2 ]
[ 548, 1990656, 2 ]
[ 549, 1990656, 2 ]
[ 552, 1990656, 4 ]
[ 553, 2654208, 2 ]
[ 554, 2654208, 2 ]
[ 555, 2654208, 2 ]
[ 556, 2654208, 4 ]
[ 557, 2654208, 2 ]
[ 558, 2654208, 4 ]
[ 559, 2654208, 4 ]
[ 560, 3981312, 6 ]
[ 561, 3981312, 4 ]
[ 562, 3981312, 4 ]
[ 563, 5308416, 4 ]
[ 564, 7962624, 6 ]
[ 565, 7962624, 6 ]
[ 566, 7962624, 12 ]
[ 567, 15925248, 12 ]


#Now we choose groups such that |v^H|<>|H| in the list res2


gap> res2:=Filtered(result1,x->x[3]<>1);

#We choose another vector v and repeat the cycle for the remaining groups 

gap> v:=[z,z^2,0*z,z^2,z,z,z^2,z^2,0*z,0*z,z,z^2,0*z,0*z^2,0*z,0*z^2];;
gap> result2:=[];
[  ]
gap> for j in res2 do
> H:=Subgroup(GL(16,3),gens[j[1]]);;
> n:=j[2];;
> m:=Size(v^H);;
> Append(result2,[[j[1],j[2],n/m]]);
> Display([j[1],j[2],n/m]);
> od;
[ 235, 110592, 1 ]
[ 247, 110592, 1 ]
[ 254, 110592, 1 ]
[ 255, 110592, 1 ]
[ 259, 110592, 1 ]
[ 260, 110592, 1 ]
[ 271, 110592, 1 ]
[ 280, 165888, 1 ]
[ 344, 221184, 1 ]
[ 352, 221184, 1 ]
[ 357, 221184, 1 ]
[ 359, 221184, 1 ]
[ 360, 221184, 1 ]
[ 361, 221184, 1 ]
[ 364, 331776, 1 ]
[ 365, 331776, 1 ]
[ 367, 331776, 2 ]
[ 368, 331776, 1 ]
[ 369, 331776, 2 ]
[ 379, 331776, 1 ]
[ 380, 331776, 1 ]
[ 381, 331776, 1 ]
[ 382, 331776, 1 ]
[ 385, 331776, 1 ]
[ 386, 331776, 1 ]
[ 392, 331776, 1 ]
[ 393, 331776, 1 ]
[ 414, 331776, 1 ]
[ 419, 331776, 1 ]
[ 420, 331776, 1 ]
[ 424, 331776, 1 ]
[ 425, 331776, 1 ]
[ 449, 442368, 1 ]
[ 451, 663552, 1 ]
[ 452, 663552, 2 ]
[ 453, 663552, 2 ]
[ 454, 663552, 1 ]
[ 455, 663552, 1 ]
[ 456, 663552, 2 ]
[ 457, 663552, 1 ]
[ 458, 663552, 2 ]
[ 461, 663552, 2 ]
[ 462, 663552, 1 ]
[ 463, 663552, 1 ]
[ 464, 663552, 1 ]
[ 465, 663552, 2 ]
[ 466, 663552, 2 ]
[ 467, 663552, 1 ]
[ 468, 663552, 2 ]
[ 469, 663552, 1 ]
[ 470, 663552, 1 ]
[ 471, 663552, 1 ]
[ 472, 663552, 1 ]
[ 473, 663552, 1 ]
[ 474, 663552, 1 ]
[ 475, 663552, 1 ]
[ 476, 663552, 1 ]
[ 477, 663552, 1 ]
[ 478, 663552, 2 ]
[ 479, 663552, 2 ]
[ 482, 663552, 2 ]
[ 483, 663552, 1 ]
[ 484, 663552, 1 ]
[ 485, 663552, 1 ]
[ 488, 663552, 2 ]
[ 494, 663552, 2 ]
[ 495, 663552, 1 ]
[ 497, 663552, 1 ]
[ 498, 663552, 1 ]
[ 499, 663552, 1 ]
[ 501, 663552, 1 ]
[ 504, 663552, 1 ]
[ 508, 995328, 1 ]
[ 512, 995328, 2 ]
[ 513, 995328, 1 ]
[ 518, 995328, 1 ]
[ 521, 1327104, 4 ]
[ 522, 1327104, 2 ]
[ 523, 1327104, 2 ]
[ 524, 1327104, 2 ]
[ 525, 1327104, 2 ]
[ 526, 1327104, 2 ]
[ 527, 1327104, 2 ]
[ 528, 1327104, 2 ]
[ 529, 1327104, 1 ]
[ 530, 1327104, 4 ]
[ 531, 1327104, 1 ]
[ 532, 1327104, 2 ]
[ 533, 1327104, 2 ]
[ 534, 1327104, 2 ]
[ 535, 1327104, 1 ]
[ 536, 1327104, 4 ]
[ 537, 1327104, 1 ]
[ 538, 1327104, 2 ]
[ 540, 1327104, 4 ]
[ 541, 1990656, 2 ]
[ 542, 1990656, 1 ]
[ 544, 1990656, 2 ]
[ 545, 1990656, 1 ]
[ 546, 1990656, 2 ]
[ 547, 1990656, 1 ]
[ 548, 1990656, 2 ]
[ 549, 1990656, 1 ]
[ 552, 1990656, 2 ]
[ 553, 2654208, 4 ]
[ 554, 2654208, 4 ]
[ 555, 2654208, 2 ]
[ 556, 2654208, 2 ]
[ 557, 2654208, 2 ]
[ 558, 2654208, 4 ]
[ 559, 2654208, 2 ]
[ 560, 3981312, 4 ]
[ 561, 3981312, 2 ]
[ 562, 3981312, 4 ]
[ 563, 5308416, 4 ]
[ 564, 7962624, 4 ]
[ 565, 7962624, 4 ]
[ 566, 7962624, 8 ]
[ 567, 15925248, 8 ]


#Now we choose groups such that |v^H|<>|H| in the list res3


gap> res3:=Filtered(result2,x->x[3]<>1);
[ [ 367, 331776, 2 ], [ 369, 331776, 2 ], [ 452, 663552, 2 ], 
  [ 453, 663552, 2 ], [ 456, 663552, 2 ], [ 458, 663552, 2 ], 
  [ 461, 663552, 2 ], [ 465, 663552, 2 ], [ 466, 663552, 2 ], 
  [ 468, 663552, 2 ], [ 478, 663552, 2 ], [ 479, 663552, 2 ], 
  [ 482, 663552, 2 ], [ 488, 663552, 2 ], [ 494, 663552, 2 ], 
  [ 512, 995328, 2 ], [ 521, 1327104, 4 ], [ 522, 1327104, 2 ], 
  [ 523, 1327104, 2 ], [ 524, 1327104, 2 ], [ 525, 1327104, 2 ], 
  [ 526, 1327104, 2 ], [ 527, 1327104, 2 ], [ 528, 1327104, 2 ], 
  [ 530, 1327104, 4 ], [ 532, 1327104, 2 ], [ 533, 1327104, 2 ], 
  [ 534, 1327104, 2 ], [ 536, 1327104, 4 ], [ 538, 1327104, 2 ], 
  [ 540, 1327104, 4 ], [ 541, 1990656, 2 ], [ 544, 1990656, 2 ], 
  [ 546, 1990656, 2 ], [ 548, 1990656, 2 ], [ 552, 1990656, 2 ], 
  [ 553, 2654208, 4 ], [ 554, 2654208, 4 ], [ 555, 2654208, 2 ], 
  [ 556, 2654208, 2 ], [ 557, 2654208, 2 ], [ 558, 2654208, 4 ], 
  [ 559, 2654208, 2 ], [ 560, 3981312, 4 ], [ 561, 3981312, 2 ], 
  [ 562, 3981312, 4 ], [ 563, 5308416, 4 ], [ 564, 7962624, 4 ], 
  [ 565, 7962624, 4 ], [ 566, 7962624, 8 ], [ 567, 15925248, 8 ] ]
  

#We choose another vector v and repeat the cycle for the remaining groups  
  
gap> v:=[z,z^2,0*z,z^2,0*z,z,z^2,z^2,0*z,0*z,z,z^2,0*z,0*z^2,0*z,0*z^2];;
gap> result3:=[];;
gap> for j in res3 do
> H:=Subgroup(GL(16,3),gens[j[1]]);;
> n:=j[2];;
> m:=Size(v^H);;
> Append(result3,[[j[1],j[2],n/m]]);
> Display([j[1],j[2],n/m]);
> od;
[ 367, 331776, 1 ]
[ 369, 331776, 1 ]
[ 452, 663552, 1 ]
[ 453, 663552, 1 ]
[ 456, 663552, 1 ]
[ 458, 663552, 1 ]
[ 461, 663552, 1 ]
[ 465, 663552, 1 ]
[ 466, 663552, 1 ]
[ 468, 663552, 1 ]
[ 478, 663552, 1 ]
[ 479, 663552, 1 ]
[ 482, 663552, 4 ]
[ 488, 663552, 1 ]
[ 494, 663552, 1 ]
[ 512, 995328, 1 ]
[ 521, 1327104, 2 ]
[ 522, 1327104, 1 ]
[ 523, 1327104, 1 ]
[ 524, 1327104, 1 ]
[ 525, 1327104, 1 ]
[ 526, 1327104, 1 ]
[ 527, 1327104, 1 ]
[ 528, 1327104, 1 ]
[ 530, 1327104, 2 ]
[ 532, 1327104, 1 ]
[ 533, 1327104, 2 ]
[ 534, 1327104, 2 ]
[ 536, 1327104, 2 ]
[ 538, 1327104, 1 ]
[ 540, 1327104, 4 ]
[ 541, 1990656, 2 ]
[ 544, 1990656, 4 ]
[ 546, 1990656, 1 ]
[ 548, 1990656, 1 ]
[ 552, 1990656, 2 ]
[ 553, 2654208, 2 ]
[ 554, 2654208, 2 ]
[ 555, 2654208, 1 ]
[ 556, 2654208, 1 ]
[ 557, 2654208, 1 ]
[ 558, 2654208, 2 ]
[ 559, 2654208, 1 ]
[ 560, 3981312, 2 ]
[ 561, 3981312, 2 ]
[ 562, 3981312, 4 ]
[ 563, 5308416, 2 ]
[ 564, 7962624, 2 ]
[ 565, 7962624, 2 ]
[ 566, 7962624, 4 ]
[ 567, 15925248, 4 ]


#Now we choose groups such that |v^H|<>|H| in the list res4


gap> res4:=Filtered(result3,x->x[3]<>1);
[ [ 482, 663552, 4 ], [ 521, 1327104, 2 ], [ 530, 1327104, 2 ], 
  [ 533, 1327104, 2 ], [ 534, 1327104, 2 ], [ 536, 1327104, 2 ], 
  [ 540, 1327104, 4 ], [ 541, 1990656, 2 ], [ 544, 1990656, 4 ], 
  [ 552, 1990656, 2 ], [ 553, 2654208, 2 ], [ 554, 2654208, 2 ], 
  [ 558, 2654208, 2 ], [ 560, 3981312, 2 ], [ 561, 3981312, 2 ], 
  [ 562, 3981312, 4 ], [ 563, 5308416, 2 ], [ 564, 7962624, 2 ], 
  [ 565, 7962624, 2 ], [ 566, 7962624, 4 ], [ 567, 15925248, 4 ] ]


# All the groups N for which |v^H|\ne |H| from the list result3 are computed to the list res4

#For the remaining groups we search for vectors v such that H on v^H is 2-closed.
#Now we define a function which check, if the $2$-closure of a group $g$ aacting on $v^g$ is equal to $g$, so that Lemma 4.4 can be applied. This function uses
#COCO2P package

gap> twoc:=function(g,v) 
>    local h,a;
>    h:=Action(g,v^g,OnRight);
>    a:=AutomorphismGroup(ColorGraph(h));
>    return IsSubgroup(a,h) and IsSubgroup(h,a); 
> end;
function( g, v ) ... end


#We check that Lemma 4.4 is satisfied for all g in the list l2, i.e. that every subgroup g in the  list l2 is 2-closed on v^g.

gap> v:=[z^3,0*z^2,0*z,0*z^2,0*z,0*z,0*z^2,0*z^2,0*z,0*z,0*z,0*z^2,0*z,0*z^2,0*z,0*z^2];;
gap> result4:=[];;
gap> for j in res4 do
> H:=Subgroup(GL(16,3),gens[j[1]]);;
> tf:=twoc(H,v);;
> Append(result4,[[j[1],j[2],tf]]);;
> Display([j[1],j[2],tf]);;         
> od;
[ 482, 663552, true ]
[ 521, 1327104, true ]
[ 530, 1327104, true ]
[ 533, 1327104, true ]
[ 534, 1327104, true ]
[ 536, 1327104, true ]
[ 540, 1327104, true ]
[ 541, 1990656, true ]
[ 544, 1990656, true ]
[ 552, 1990656, true ]
[ 553, 2654208, true ]
[ 554, 2654208, true ]
[ 558, 2654208, true ]
[ 560, 3981312, false ]
[ 561, 3981312, true ]
[ 562, 3981312, true ]
[ 563, 5308416, true ]
[ 564, 7962624, false ]
[ 565, 7962624, false ]
[ 566, 7962624, true ]
[ 567, 15925248, true ]


#Now we choose groups such that the restriciton of H on v^H is not 2-closed in the list res5

res5:=Filtered(result4,x->not x[3]);
[ [ 560, 3981312, false ], [ 564, 7962624, false ], [ 565, 7962624, false ] ]

#We choose another vector v and check, if the restriction of H on v^H is 2-closed

gap> v:=[z,0*z^2,0*z,0*z^2,0*z,0*z,0*z^2,0*z^2,0*z,0*z,0*z,0*z^2,0*z,0*z^2,0*z,z^2];;
gap> result5:=[];;
gap> for j in res5 do
> H:=Subgroup(GL(16,3),gens[j[1]]);;
> tf:=twoc(H,v);;
> Append(result4,[[j[1],j[2],tf]]);;
> Display([j[1],j[2],tf]);; 
> od;
[ 560, 3981312, true ]
[ 564, 7962624, true ]
[ 565, 7962624, true ]
gap>