#We find all primitive subgroups of $GL(18,2)$ with parameters $e=9$, $a=2$
#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 $3$, $F=U\circ E$ and $E$ is extraspecial of order $3^{1+4}$, moreover
#in this case $|H:A|$ divides $2$. 

gap> e:=9;; p:=2;; d:=18;; a:=2;;

#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 $|F|=(2^2-1)*9*9$

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

#In view of Lemma 5.4, we choose subgroups satisfying $\vert A/F\vert\in \{40, 320, 192,1152\}$ so $|H:F|$ lies in 
#$\{40, 320, 192, 1152, a*40, a*320, a*192, a*1152\}$, and collect these subgroups in the list l2.


gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [40, 320, 192, 1152, a*40, a*320, a*192, a*1152]);;
gap> Size(l2);
31

#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);;

gap> v:=[z1,z0,z0,z0,z0,z0,z0,z1,z1,z0,z0,z1,z1,z1,z0,z0,z0,z0];;
gap> result:=[];; 
gap> for j in l2 do
> tf:=twoc(j,v);;
> n:=Size(j);;                   
> Append(result,[[n,tf]]);;
> Display([n,tf]);
> od;
[ 559872, true ]
[ 279936, true ]
[ 279936, true ]
[ 93312, true ]
[ 93312, true ]
[ 93312, true ]
[ 93312, true ]
[ 93312, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 19440, true ]
[ 9720, true ]
[ 9720, true ]
[ 93312, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 155520, true ]
[ 279936, true ]
[ 93312, true ]
[ 93312, true ]
[ 46656, true ]
[ 46656, true ]
[ 46656, true ]
[ 9720, true ]
[ 46656, true ]
[ 77760, true ]