#We find all primitive subgroups of $GL(6,3)$ with parameters $e=2$, $a=3$
#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 $26$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+2}$, moreover
#in this case $|H:A|$ divides 3. Since $b=2$ does not divide $a=3$, Lemma 5.3 implies that there are two nonconjugate subgroups $F_1$ and $F_2$, and
#$N_{GL(6,3)}(F_1)/F_1\simeq Q^+(2,2)$, while  $N_{GL(6,3)}(F_2)/F_2\simeq Q^-(2,2)$

gap> e:=2;; p:=3;; d:=6;; a:=3;;

#Now we generate the list of all primitive subgroups in $GL(6,3)$, using standart function of the package IRREDSOL

gap> l0:=AllIrreducibleSolvableMatrixGroups(Degree,d,Field,GF(p),IsPrimitiveMatrixGroup,true);;

#In this list we choose subgroups, whose order is divisible by the order of $F$ in the notations of Theorem 5.1

gap> l1:=Filtered(l0,x-> (Order(x) mod ((p^a-1)*e*e)) = 0);; 

#In view of Lemma 5.3, we choose subgroups satisfying $\vert A/F\vert\in \{2,6\}$ and collect these subgroups in the list l2.

gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [2,6,3*6]);; Size(l2);
4

#We choose vector v and check that the subgroup H from the list l2 acts regualrly on v^H, i.e. |H|=|v^H|, so Corollary 2.5 can be applied


gap> v:=[Z(3),Z(3),0*Z(3),Z(3),Z(3),Z(3)];;
gap> result:=[];; i:=0;; 
gap> for H in l2 do
> i:=i+1;; n:=Size(H);; m:=Size(v^H);;
> result[i]:=[i,n,n/m];
> Display(result[i]);
> od;
[ 1, 1872, 3 ]
[ 2, 624, 1 ]
[ 3, 208, 1 ]
[ 4, 624, 1 ]

#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 the first group in the list l2, i.e. that l2[1] in the  list l2 is 2-closed on v^l2[1].

gap> twoc(l2[1],v);
true