#We find all primitive subgroups of $GL(9,7)$ with parameters $e=9$, $a=1$
#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 $16$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+6}$, in this case $H=A$. 

#First we make general variables:

gap> z:=Z(17);; v:=[z,z^3,z^5,0*z,z^7,z^9,z^11,z^13];;


#Now we run the following loop, where gens is the list of generators from generators-GL(8,17).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> result:=[];; i:=0;;
gap> for k in gens do
> H:=Subgroup(GL(8,17),k);; i:=i+1;;
> n:=Size(H);; m:=Size(v^H);;
> result[i]:=[i,n,n/m];
> Display(result[i]);
> od;
[ 1, 9216, 1 ]
[ 2, 9216, 1 ]
[ 3, 14336, 1 ]
[ 4, 15360, 1 ]
[ 5, 18432, 1 ]
[ 6, 18432, 1 ]
[ 7, 18432, 1 ]
[ 8, 18432, 1 ]
[ 9, 27648, 1 ]
[ 10, 27648, 1 ]
[ 11, 27648, 1 ]
[ 12, 30720, 1 ]
[ 13, 30720, 1 ]
[ 14, 30720, 1 ]
[ 15, 43008, 1 ]
[ 16, 36864, 1 ]
[ 17, 55296, 1 ]
[ 18, 55296, 1 ]
[ 19, 55296, 1 ]
[ 20, 55296, 1 ]
[ 21, 55296, 1 ]
[ 22, 55296, 1 ]
[ 23, 55296, 1 ]
[ 24, 55296, 1 ]
[ 25, 61440, 1 ]
[ 26, 61440, 1 ]
[ 27, 61440, 1 ]
[ 28, 82944, 1 ]
[ 29, 110592, 1 ]
[ 30, 110592, 1 ]
[ 31, 110592, 1 ]
[ 32, 110592, 1 ]
[ 33, 110592, 1 ]
[ 34, 110592, 1 ]
[ 35, 110592, 1 ]
[ 36, 110592, 1 ]
[ 37, 110592, 1 ]
[ 38, 110592, 2 ]
[ 39, 110592, 1 ]
[ 40, 122880, 1 ]
[ 41, 165888, 1 ]
[ 42, 165888, 1 ]
[ 43, 165888, 1 ]
[ 44, 221184, 1 ]
[ 45, 221184, 1 ]
[ 46, 221184, 1 ]
[ 47, 221184, 2 ]
[ 48, 221184, 1 ]
[ 49, 221184, 1 ]
[ 50, 221184, 1 ]
[ 51, 221184, 2 ]
[ 52, 221184, 1 ]
[ 53, 221184, 1 ]
[ 54, 331776, 1 ]
[ 55, 331776, 1 ]
[ 56, 442368, 1 ]
[ 57, 442368, 2 ]
[ 58, 663552, 1 ]
[ 59, 663552, 1 ]
[ 60, 663552, 1 ]
[ 61, 663552, 1 ]
[ 62, 1327104, 2 ]
[ 63, 1327104, 1 ]


#Now we choose groups such that |v^H|<>|H| in the list res1


gap> res1:=Filtered(result,x->x[3]<>1);; Size(res1);
5

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


gap> v:=[7*z,z^3,z^5,3*z,0*z^7,0*z^9,0*z^11,6*z^13];;
gap> result1:=[];; 
gap> for k in res1 do
> H:=Subgroup(GL(8,17),gens[k[1]]);; 
> n:=k[2];; m:=Size(v^H);;
> Append(result1,[[k[1],n,n/m]]);
> Display([k[1],n,n/m]);
> od;
[ 38, 110592, 1 ]
[ 47, 221184, 1 ]
[ 51, 221184, 1 ]
[ 57, 442368, 1 ]
[ 62, 1327104, 1 ]