#We find all primitive subgroups of $GL(4,59)$ with parameters $e=4$, $a=1$. 
#In the notations of Theorem 5.1, if $H\leq GL(4,59)$ is a primitive solvable subgroup, then it possesses a series

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

#where $U$ is cyclic of order $58$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+4}$, moreover
#in this case $|H:A|$ equals $1$. Since $b=2$ does not divide $a=1$, Lemma 5.3 implies that there are two nonconjugate subgroups $F_1$ and $F_2$, and
#$N_{GL(4,59)}(F_1)/F_1\simeq Q^+(4,2)$, while  $N_{GL(4,59)}(F_2)/F_2\simeq Q^-(4,2)$


#First we generate the list of all primitive subgroups in $GL(4,59)$, using standart function of the package IRREDSOL

gap> e:=4;; p:=59;; d:=4;; a:=1;;

#Now we generate the list of all primitive subgroups in $GL(4,59)$, 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 $\vert A/F\vert\in \{12,20,72\}$ and collect these subgroups in the list l2.

gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [12,20,72]);;

#Now we run the following loop. 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> v:=[Z(59),0*Z(59),0*Z(59),Z(59)];;
gap> result:=[];; i:=0;;
gap> for j in l2 do
> n:=Size(j);; m:=Size(v^j);; i:=i+1;; Append(result,[[i,n,n/m]]);;
> Display([i,n,n/m]);
> od;
[ 1, 11136, 1 ]
[ 2, 66816, 48 ]
[ 3, 18560, 4 ]

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


gap> res1:=Filtered(result,x->x[3]<>1);; Size(res1);    
2

#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> for j in res1 do
> tf:=twoc(l2[j[1]],v);; n:=j[2];; i:=i+1;; Append(result1,[[j[1],n,tf]]);;
> Display([j[1],n,tf]);
> od;
[ 2, 66816, true ]
[ 3, 18560, true ]