#We find all primitive subgroups of $GL(2,3)$ with parameters $e=2$, $a=1$

gap> e:=2;; p:=3;; d:=2;; a:=1;;

#Now we generate the list of all primitive subgroups in $GL(2,3)$, 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 6.1

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

#In view of Lemma 6.3, 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]);;

#We define the vector $v$ and check that all groups in l2 are transitive on the nonzero vectors

gap> v:=[Z(3),Z(3)^2];;
gap> so:=List(l2,x->[Size(v^x),Size(x)]);
[ [ 8, 48 ], [ 8, 16 ] ]
gap>