#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(GL(16,3),gens[j]);;
> n:=order[j];
> m:=Size(v^H);
> result[j]:=[j,n,n/m];
> od;
gap> result;
[ [ 1, 512, 1 ], [ 2, 1024, 1 ], [ 3, 1024, 1 ], 
  [ 4, 1024, 1 ], [ 5, 8704, 1 ], [ 6, 7168, 1 ], 
  [ 7, 7168, 1 ], [ 8, 7680, 1 ], [ 9, 17408, 1 ], 
  [ 10, 9216, 1 ], [ 11, 9216, 1 ], [ 12, 9216, 1 ], 
  [ 13, 9216, 1 ], [ 14, 9216, 1 ], [ 15, 9216, 1 ], 
  [ 16, 14336, 1 ], [ 17, 15360, 1 ], [ 18, 15360, 1 ], 
  [ 19, 15360, 1 ], [ 20, 15360, 1 ], [ 21, 15360, 1 ], 
  [ 22, 15360, 1 ], [ 23, 15360, 1 ], [ 24, 21504, 1 ], 
  [ 25, 21504, 1 ], [ 26, 21504, 1 ], [ 27, 21504, 1 ], 
  [ 28, 21504, 1 ], [ 29, 23040, 1 ], [ 30, 34816, 1 ], 
  [ 31, 18432, 2 ], [ 32, 18432, 2 ], [ 33, 18432, 1 ], 
  [ 34, 18432, 1 ], [ 35, 18432, 1 ], [ 36, 18432, 1 ], 
  [ 37, 18432, 1 ], [ 38, 27648, 1 ], [ 39, 27648, 1 ], 
  [ 40, 27648, 1 ], [ 41, 27648, 1 ], [ 42, 27648, 1 ], 
  [ 43, 27648, 1 ], [ 44, 27648, 1 ], [ 45, 27648, 1 ], 
  [ 46, 27648, 1 ], [ 47, 27648, 1 ], [ 48, 27648, 1 ], 
  [ 49, 30720, 1 ], [ 50, 30720, 1 ], [ 51, 30720, 1 ], 
  [ 52, 30720, 1 ], [ 53, 30720, 1 ], [ 54, 30720, 1 ], 
  [ 55, 30720, 1 ], [ 56, 30720, 1 ], [ 57, 30720, 1 ], 
  [ 58, 30720, 1 ], [ 59, 30720, 1 ], [ 60, 43008, 1 ], 
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  [ 64, 46080, 1 ], [ 65, 46080, 1 ], [ 66, 46080, 1 ], 
  [ 67, 46080, 1 ], [ 68, 46080, 2 ], [ 69, 64512, 1 ], 
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  [ 76, 55296, 2 ], [ 77, 55296, 1 ], [ 78, 55296, 1 ], 
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  [ 82, 55296, 2 ], [ 83, 55296, 1 ], [ 84, 55296, 2 ], 
  [ 85, 55296, 1 ], [ 86, 55296, 1 ], [ 87, 55296, 1 ], 
  [ 88, 55296, 2 ], [ 89, 55296, 1 ], [ 90, 55296, 1 ], 
  [ 91, 55296, 1 ], [ 92, 55296, 1 ], [ 93, 55296, 2 ], 
  [ 94, 55296, 2 ], [ 95, 55296, 1 ], [ 96, 55296, 1 ], 
  [ 97, 55296, 1 ], [ 98, 55296, 2 ], [ 99, 55296, 1 ], 
  [ 100, 55296, 1 ], [ 101, 55296, 1 ], [ 102, 61440, 1 ], 
  [ 103, 61440, 1 ], [ 104, 61440, 1 ], [ 105, 61440, 1 ], 
  [ 106, 61440, 1 ], [ 107, 61440, 1 ], [ 108, 61440, 1 ], 
  [ 109, 82944, 1 ], [ 110, 82944, 1 ], [ 111, 82944, 1 ], 
  [ 112, 82944, 1 ], [ 113, 92160, 1 ], [ 114, 92160, 1 ], 
  [ 115, 92160, 1 ], [ 116, 92160, 1 ], [ 117, 92160, 1 ], 
  [ 118, 92160, 1 ], [ 119, 92160, 2 ], [ 120, 92160, 1 ], 
  [ 121, 92160, 1 ], [ 122, 92160, 1 ], [ 123, 92160, 1 ], 
  [ 124, 92160, 2 ], [ 125, 92160, 2 ], [ 126, 92160, 1 ], 
  [ 127, 92160, 1 ], [ 128, 92160, 1 ], [ 129, 129024, 1 ], 
  [ 130, 110592, 2 ], [ 131, 110592, 1 ], [ 132, 110592, 1 ], 
  [ 133, 110592, 1 ], [ 134, 110592, 2 ], [ 135, 110592, 2 ], 
  [ 136, 110592, 2 ], [ 137, 110592, 4 ], [ 138, 110592, 2 ], 
  [ 139, 110592, 2 ], [ 140, 110592, 1 ], [ 141, 110592, 1 ], 
  [ 142, 110592, 2 ], [ 143, 110592, 1 ], [ 144, 110592, 1 ], 
  [ 145, 110592, 1 ], [ 146, 110592, 1 ], [ 147, 110592, 1 ], 
  [ 148, 110592, 1 ], [ 149, 110592, 1 ], [ 150, 110592, 1 ], 
  [ 151, 110592, 1 ], [ 152, 110592, 1 ], [ 153, 110592, 2 ], 
  [ 154, 110592, 2 ], [ 155, 110592, 2 ], [ 156, 110592, 1 ], 
  [ 157, 110592, 1 ], [ 158, 110592, 1 ], [ 159, 110592, 1 ], 
  [ 160, 110592, 2 ], [ 161, 110592, 1 ], [ 162, 110592, 2 ], 
  [ 163, 110592, 2 ], [ 164, 110592, 2 ], [ 165, 110592, 2 ], 
  [ 166, 110592, 2 ], [ 167, 110592, 2 ], [ 168, 122880, 1 ], 
  [ 169, 165888, 1 ], [ 170, 165888, 2 ], [ 171, 165888, 1 ], 
  [ 172, 165888, 2 ], [ 173, 165888, 2 ], [ 174, 165888, 1 ], 
  [ 175, 184320, 1 ], [ 176, 184320, 1 ], [ 177, 184320, 1 ], 
  [ 178, 184320, 1 ], [ 179, 184320, 2 ], [ 180, 184320, 2 ], 
  [ 181, 184320, 2 ], [ 182, 184320, 1 ], [ 183, 184320, 1 ], 
  [ 184, 184320, 1 ], [ 185, 184320, 1 ], [ 186, 184320, 2 ], 
  [ 187, 184320, 1 ], [ 188, 184320, 1 ], [ 189, 184320, 2 ], 
  [ 190, 221184, 2 ], [ 191, 221184, 2 ], [ 192, 221184, 1 ], 
  [ 193, 221184, 1 ], [ 194, 221184, 2 ], [ 195, 221184, 2 ], 
  [ 196, 221184, 1 ], [ 197, 221184, 4 ], [ 198, 221184, 2 ], 
  [ 199, 221184, 2 ], [ 200, 221184, 2 ], [ 201, 221184, 2 ], 
  [ 202, 221184, 2 ], [ 203, 221184, 4 ], [ 204, 221184, 4 ], 
  [ 205, 221184, 2 ], [ 206, 221184, 1 ], [ 207, 221184, 2 ],   
  [ 208, 221184, 1 ], [ 209, 221184, 2 ], [ 210, 331776, 1 ], 
  [ 211, 331776, 1 ], [ 212, 331776, 1 ], [ 213, 331776, 1 ], 
  [ 214, 331776, 2 ], [ 215, 368640, 1 ], [ 216, 368640, 2 ], 
  [ 217, 368640, 2 ], [ 218, 368640, 2 ], [ 219, 368640, 1 ], 
  [ 220, 368640, 1 ], [ 221, 368640, 1 ], [ 222, 442368, 4 ], 
  [ 223, 442368, 2 ], [ 224, 663552, 1 ], [ 225, 663552, 2 ], 
  [ 226, 663552, 2 ], [ 227, 663552, 2 ], [ 228, 663552, 2 ], 
  [ 229, 663552, 2 ], [ 230, 663552, 2 ], [ 231, 663552, 2 ], 
  [ 232, 737280, 2 ], [ 233, 1327104, 2 ], 
  [ 234, 1327104, 4 ] ]
  
  
  #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,z,z^2,z,z,z^2,z^2,0*z,z,z,z^2,0*z,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;
[ 31, 18432, 1 ]
[ 32, 18432, 1 ]
[ 68, 46080, 1 ]
[ 72, 36864, 1 ]
[ 73, 55296, 1 ]
[ 74, 55296, 1 ]
[ 76, 55296, 1 ]
[ 80, 55296, 1 ]
[ 82, 55296, 1 ]
[ 84, 55296, 1 ]
[ 88, 55296, 1 ]
[ 93, 55296, 1 ]
[ 94, 55296, 1 ]
[ 98, 55296, 1 ]
[ 119, 92160, 1 ]
[ 124, 92160, 1 ]
[ 125, 92160, 1 ]
[ 130, 110592, 1 ]
[ 134, 110592, 1 ]
[ 135, 110592, 1 ]
[ 136, 110592, 1 ]
[ 137, 110592, 1 ]
[ 138, 110592, 1 ]
[ 139, 110592, 1 ]
[ 142, 110592, 1 ]
[ 153, 110592, 1 ]
[ 154, 110592, 1 ]
[ 155, 110592, 1 ]
[ 160, 110592, 1 ]
[ 162, 110592, 1 ]
[ 163, 110592, 1 ]
[ 164, 110592, 1 ]
[ 165, 110592, 1 ]
[ 166, 110592, 1 ]
[ 167, 110592, 1 ]
[ 170, 165888, 1 ]
[ 172, 165888, 1 ]
[ 173, 165888, 1 ]
[ 179, 184320, 1 ]
[ 180, 184320, 1 ]
[ 181, 184320, 1 ]
[ 186, 184320, 1 ]
[ 189, 184320, 1 ]
[ 190, 221184, 1 ]
[ 191, 221184, 1 ]
[ 194, 221184, 1 ]
[ 195, 221184, 1 ]
[ 197, 221184, 1 ]
[ 198, 221184, 1 ]
[ 199, 221184, 1 ]
[ 200, 221184, 1 ]
[ 201, 221184, 1 ]
[ 202, 221184, 1 ]
[ 203, 221184, 1 ]
[ 204, 221184, 1 ]
[ 205, 221184, 1 ]
[ 207, 221184, 1 ]
[ 209, 221184, 1 ]
[ 214, 331776, 1 ]
[ 216, 368640, 1 ]
[ 217, 368640, 1 ]
[ 218, 368640, 1 ]
[ 222, 442368, 1 ]
[ 223, 442368, 1 ]
[ 225, 663552, 1 ]
[ 226, 663552, 1 ]
[ 227, 663552, 1 ]
[ 228, 663552, 1 ]
[ 229, 663552, 1 ]
[ 230, 663552, 2 ]
[ 231, 663552, 1 ]
[ 232, 737280, 1 ]
[ 233, 1327104, 1 ]
[ 234, 1327104, 2 ]



#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:=[0*z,0*z^2,z,z,0*z,z,z,z,z,z,z,z^2,z,0*z^2,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;
[ 230, 663552, 1 ]
[ 234, 1327104, 1 ]