#We find all primitive subgroups of $GL(8,3)$ with parameters $e=2$, $a=4$
#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 $80$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+2}$, moreover
#in this case $|H:A|$ divides $4$. Since $b=2$ divides $a=4$, Lemma 5.3 implies that up to conjugation there is one $F$, and
#$N_{L}(F)/F\simeq Sp(2,2)\simeq S_3$, where $L=C_{GL(8,3)}(U)$.

gap> e:=2;; p:=3;; d:=8;; a:=4;;

#Now we generate the list of all primitive subgroups in $GL(8,3)$, 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 $\vert A/F\vert\in \{6\}$ and collect these subgroups in the list l2.

gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [6,2*6,4*6]);; Size(l2);
30

#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> v:=[Z(3),Z(3),0*Z(3),Z(3),Z(3),Z(3),Z(3),Z(3)];;
gap> result:=[];; i:=0;;
gap> for j in l2 do
> tf:=twoc(j,v);; i:=i+1;; n:=Size(j);;                   
> Append(result,[[i,n,tf]]);;
> Display([i,n,tf]);
> od;
[ 1, 7680, true ]
[ 2, 3840, true ]
[ 3, 3840, true ]
[ 4, 1920, true ]
[ 5, 1920, true ]
[ 6, 3840, true ]
[ 7, 3840, true ]
[ 8, 3840, true ]
[ 9, 3840, true ]
[ 10, 1920, true ]
[ 11, 1920, true ]
[ 12, 1920, true ]
[ 13, 1920, true ]
[ 14, 1920, true ]
[ 15, 1920, true ]
[ 16, 7680, true ]
[ 17, 3840, true ]
[ 18, 1920, true ]
[ 19, 3840, true ]
[ 20, 3840, true ]
[ 21, 7680, true ]
[ 22, 7680, true ]
[ 23, 3840, true ]
[ 24, 1920, true ]
[ 25, 1920, true ]
[ 26, 1920, true ]
[ 27, 1920, true ]
[ 28, 1920, true ]
[ 29, 1920, true ]
[ 30, 1920, true ]