#We find all primitive subgroups of $GL(9,13)$ 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 $12$, $F=U\circ E$ and $E$ is extraspecial of order $3^{1+4}$, in this case $H=A$. 

#First we make general variables:

gap> z:=Z(13);;
gap> v:=[z,z^5,z^2,0*z,z^7,z^4,z^3,z^11,z^8];;
gap> result:=[];;

#Now we run the following loop, where gens is the list of generators from generators-GL(9,13).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(9,13),gens[j]);;
> n:=Size(H);;
> m:=Size(v^H);
> result[j]:=[j,n,n/m];
> Display(result[j]);
> od;
[ 1, 4860, 1 ]
[ 2, 9720, 1 ]
[ 3, 7776, 1 ]
[ 4, 7776, 1 ]
[ 5, 19440, 1 ]
[ 6, 15552, 1 ]
[ 7, 15552, 1 ]
[ 8, 15552, 1 ]
[ 9, 15552, 1 ]
[ 10, 15552, 1 ]
[ 11, 15552, 1 ]
[ 12, 15552, 1 ]
[ 13, 23328, 1 ]
[ 14, 38880, 1 ]
[ 15, 31104, 1 ]
[ 16, 31104, 1 ]
[ 17, 31104, 1 ]
[ 18, 31104, 1 ]
[ 19, 31104, 1 ]
[ 20, 31104, 1 ]
[ 21, 46656, 1 ]
[ 22, 46656, 1 ]
[ 23, 46656, 1 ]
[ 24, 46656, 1 ]
[ 25, 46656, 1 ]
[ 26, 62208, 1 ]
[ 27, 62208, 1 ]
[ 28, 62208, 1 ]
[ 29, 93312, 1 ]
[ 30, 93312, 1 ]
[ 31, 93312, 1 ]
[ 32, 93312, 1 ]
[ 33, 93312, 1 ]
[ 34, 155520, 1 ]
[ 35, 124416, 1 ]
[ 36, 186624, 1 ]
[ 37, 186624, 1 ]
[ 38, 186624, 1 ]
[ 39, 186624, 1 ]
[ 40, 311040, 1 ]
[ 41, 373248, 1 ]
[ 42, 373248, 1 ]
[ 43, 559872, 1 ]
[ 44, 1119744, 1 ]