#We find all primitive subgroups of $GL(8,5)$ with parameters $e=4$, $a=2$. 
#In the notations of Theorem 5.1, if $H\leq GL(8,5)$ is a primitive solvable subgroup, then it possesses a series

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

#where $U$ is cyclic of order $24$, $F=U\circ E$ and $E$ is extraspecial of order $2^{1+4}$, moreover
#in this case $|H:A|$ divides $2$. Since $b=1$ divides $a=2$, Lemma 5.3 implies that there is one up to conjugation subgroup $F$, and
#$N_{GL(8,5)}(F)/F\simeq Sp(4,2)$.

gap> e:=4;; p:=5;; d:=8;; a:=2;;

#Now we generate the list of all primitive subgroups in $GL(8,5)$, 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 $\vert A/F\vert\in \{20,72\}$ and collect these subgroups in the list l2.

gap> l2:=Filtered(l1,x-> (Order(x)/((p^a-1)*e*e)) in  [20,72, 2*20,2*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> z:=Z(5);; v:=[z,0*z,z^2,z^4,0*z,0*z,z,0*z];;
gap> result:=[];;
gap> 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, 55296, 2 ]
[ 2, 27648, 2 ]
[ 3, 27648, 1 ]
[ 4, 27648, 1 ]
[ 5, 27648, 2 ]
[ 6, 27648, 2 ]
[ 7, 27648, 1 ]
[ 8, 27648, 1 ]
[ 9, 27648, 2 ]
[ 10, 27648, 2 ]
[ 11, 27648, 2 ]
[ 12, 27648, 1 ]
[ 13, 27648, 1 ]
[ 14, 27648, 1 ]
[ 15, 27648, 1 ]
[ 16, 15360, 1 ]
[ 17, 7680, 1 ]
[ 18, 7680, 1 ]
[ 19, 7680, 1 ]
[ 20, 7680, 1 ]
[ 21, 7680, 1 ]
[ 22, 7680, 1 ]
[ 23, 27648, 1 ]
[ 24, 55296, 2 ]
[ 25, 55296, 1 ]
[ 26, 55296, 1 ]
[ 27, 27648, 2 ]
[ 28, 27648, 1 ]
[ 29, 27648, 2 ]
[ 30, 27648, 2 ]
[ 31, 27648, 1 ]
[ 32, 27648, 1 ]
[ 33, 27648, 1 ]
[ 34, 27648, 1 ]
[ 35, 55296, 2 ]
[ 36, 55296, 2 ]
[ 37, 55296, 1 ]
[ 38, 55296, 1 ]
[ 39, 27648, 2 ]
[ 40, 27648, 2 ]
[ 41, 27648, 1 ]
[ 42, 27648, 1 ]
[ 43, 15360, 1 ]
[ 44, 7680, 1 ]
[ 45, 7680, 1 ]
[ 46, 7680, 1 ]
[ 47, 15360, 1 ]
[ 48, 15360, 1 ]
[ 49, 7680, 1 ]
[ 50, 7680, 1 ]
[ 51, 7680, 1 ]
[ 52, 7680, 1 ]
[ 53, 27648, 2 ]
[ 54, 55296, 2 ]
[ 55, 27648, 2 ]
[ 56, 55296, 2 ]
[ 57, 55296, 2 ]
[ 58, 27648, 1 ]
[ 59, 27648, 1 ]
[ 60, 27648, 1 ]
[ 61, 27648, 2 ]
[ 62, 7680, 1 ]


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


gap> res1:=Filtered(result,x->x[3]<>1);;              
gap> Size(res1);
21

#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 define the vector $v$.
#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,0*z,z^2,z^4,0*z,0*z,0*z,0*z];;
gap> result1:=[];; 
gap> for j in res1 do
> tf:=twoc(l2[j[1]],v);;                   
> Append(result,[[j[1],j[2],tf]]);;
> Display([j[1],j[2],tf]);
> od;
[ 1, 55296, true ]
[ 2, 27648, true ]
[ 5, 27648, true ]
[ 6, 27648, true ]
[ 9, 27648, true ]
[ 10, 27648, true ]
[ 11, 27648, true ]
[ 24, 55296, true ]
[ 27, 27648, true ]
[ 29, 27648, true ]
[ 30, 27648, true ]
[ 35, 55296, true ]
[ 36, 55296, true ]
[ 39, 27648, true ]
[ 40, 27648, true ]
[ 53, 27648, true ]
[ 54, 55296, true ]
[ 55, 27648, true ]
[ 56, 55296, true ]
[ 57, 55296, true ]
[ 61, 27648, true ]