#We find all primitive subgroups of $GL(18,2)$ with parameters $e=3$, $a=6$
#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 $63$, $F=U\circ E$ and $E$ is extraspecial of order $3^{1+2}$, moreover
#in this case $|H:A|$ divides $6$.

gap> e:=3;; p:=2;; d:=18;; a:=6;;

#Now we generate the list of all primitive subgroups in $GL(18,2)$, 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 $\vert A/F\vert\in \{24\}$ and collect these subgroups in the list l2.

gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [24,2*24,3*24,6*24]);; Size(l2);


#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> z1:=Z(2);; z0:=0*Z(2);; v:=[z0,z0,z1,z1,z0,z0,z0,z1,z1,z1,z1,z1,z1,z1,z1,z0,z1,z0];;
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, 27216, 3 ]
[ 2, 81648, 1 ]
[ 3, 27216, 1 ]
[ 4, 27216, 1 ]
[ 5, 27216, 1 ]
[ 6, 13608, 1 ]
[ 7, 13608, 1 ]
[ 8, 40824, 2 ]
[ 9, 13608, 2 ]
[ 10, 13608, 2 ]
[ 11, 13608, 2 ]
[ 12, 13608, 2 ]
[ 13, 13608, 2 ]
[ 14, 13608, 2 ]
[ 15, 13608, 2 ]
[ 16, 13608, 2 ]
[ 17, 27216, 6 ]
[ 18, 13608, 1 ]


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

gap> res1:=Filtered(result,x->x[3]<>1);; Size(res1);
11

#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> result1:=[];; i:=0;;
gap> for j in res1 do
> H:=l2[j[1]];; tf:=twoc(H,v);; n:=j[2];; i:=i+1;;                   
> Append(result1,[[j[1],n,tf]]);;
> Display([j[1],n,tf]);
> od;
[ 1, 27216, true ]
[ 8, 40824, true ]
[ 9, 13608, true ]
[ 10, 13608, true ]
[ 11, 13608, true ]
[ 12, 13608, true ]
[ 13, 13608, true ]
[ 14, 13608, true ]
[ 15, 13608, true ]
[ 16, 13608, true ]
[ 17, 27216, true ]