#We find all primitive subgroups of $GL(18,5)$ with parameters $e=9$, $a=2$. In this case $b=2$ divides $a$, so we use item (i) of Lemma 5.5. We start with
#primitive solvable subgroups in $GL(9,5^b)=GL(9,25)$ and then produce primitive subgroups in $GL(18,5)$.
#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 $24$, $F=U\circ E$ and $E$ is extraspecial of order $3^{1+4}$, moreover
#in this case $|H:A|$ divides $2$. 

gap> e:=9;; p:=5;; d:=18;; a:=2;;

#Since 5-1 is not divisible by 3, we take generators of primitive subgroups in $GL(9,25)$ from the file generators-GL(9,25).txt as a list gens for every matrix 
#g in gens we substitute each entry $g_{i,j}\in GF(5^2)$ by a 2\times2-matrix over $GF(5)$


gap> bas:= Basis( AsVectorSpace( GF(p),GF(p^a) ) );;
gap> genS:=[];;                               
gap> for i in [1..Size(gens)] do
> genS[i]:=[];;
> for j in gens[i]do             
> Append(genS[i],[BlownUpMat(bas,j)]);;           
> od;
> od;

#Now we produce Singer cycle in $GL(2,5)$ that agrees with generators obtained above and element from normalizer inducing the field automorphism of order $2$. 
#We find a central element in subgroup H with generators genS[44], its center is cyclic of order $24$.

gap> H:=Subgroup(GL(18,5),genS[44]);;
gap> Zz:=Center(H);
<group of 18x18 matrices over GF(5)>
gap> Order(Zz);
24
gap> gn:=GeneratorsOfGroup(Zz);
[ < immutable compressed matrix 18x18 over GF(5) >, 
  < immutable compressed matrix 18x18 over GF(5) >, 
  < immutable compressed matrix 18x18 over GF(5) >, 
  < immutable compressed matrix 18x18 over GF(5) > ]
gap> Order(gn[1]);
12
gap> Order(gn[2]);
8
gap> Order(gn[3]);
12
gap> Order(gn[4]);
8
gap> x:=gn[1]^4*gn[2];;
gap> Order(x);
24
gap> Display(x);
 2 3 . . . . . . . . . . . . . . . .
 4 . . . . . . . . . . . . . . . . .
 . . 2 3 . . . . . . . . . . . . . .
 . . 4 . . . . . . . . . . . . . . .
 . . . . 2 3 . . . . . . . . . . . .
 . . . . 4 . . . . . . . . . . . . .
 . . . . . . 2 3 . . . . . . . . . .
 . . . . . . 4 . . . . . . . . . . .
 . . . . . . . . 2 3 . . . . . . . .
 . . . . . . . . 4 . . . . . . . . .
 . . . . . . . . . . 2 3 . . . . . .
 . . . . . . . . . . 4 . . . . . . .
 . . . . . . . . . . . . 2 3 . . . .
 . . . . . . . . . . . . 4 . . . . .
 . . . . . . . . . . . . . . 2 3 . .
 . . . . . . . . . . . . . . 4 . . .
 . . . . . . . . . . . . . . . . 2 3
 . . . . . . . . . . . . . . . . 4 .


#Now we produce t so that $I_9\otimes t=x$ and find $s$ such that $t^s=t^5$

gap> z:=Z(5);;
gap> t:=[[z,z^(-1)],[z^2,0*z]];;
gap> Display(t);                
 2 3
 4 .
gap> s:=RepresentativeAction(GL(a,p),t,t^p);;
gap> s18:=KroneckerProduct(IdentityMat(9,Z(5)),s);;
gap> Display(s18);                                 
 4 3 . . . . . . . . . . . . . . . .
 . 1 . . . . . . . . . . . . . . . .
 . . 4 3 . . . . . . . . . . . . . .
 . . . 1 . . . . . . . . . . . . . .
 . . . . 4 3 . . . . . . . . . . . .
 . . . . . 1 . . . . . . . . . . . .
 . . . . . . 4 3 . . . . . . . . . .
 . . . . . . . 1 . . . . . . . . . .
 . . . . . . . . 4 3 . . . . . . . .
 . . . . . . . . . 1 . . . . . . . .
 . . . . . . . . . . 4 3 . . . . . .
 . . . . . . . . . . . 1 . . . . . .
 . . . . . . . . . . . . 4 3 . . . .
 . . . . . . . . . . . . . 1 . . . .
 . . . . . . . . . . . . . . 4 3 . .
 . . . . . . . . . . . . . . . 1 . .
 . . . . . . . . . . . . . . . . 4 3
 . . . . . . . . . . . . . . . . . 1

#We check, that s18 normalizes Fitting subgroup of H

gap> F:=FittingSubgroup(H);                                           
<group of 18x18 matrices over GF(5)>
gap> Order(F);
1944
gap> F^s18=F;
true


#Now we find the full normalizer of $F$ in $GL(18,5)$ isomorphic to $((24\circ  3^{1+4}).Sp(4,3)).2$ and has order 
#$(p^a-1)*e^2*|Sp(4,3)|*2=(5^2-1)*9^2*3^4*(3^2-1)*(3^4-1)*2$

gap> gen:=Concatenation(genS[44],[s18]);;    
gap> N:=Subgroup(GL(18,5),gen);;
gap> n:=Size(N);
201553920
gap> n=(5^2-1)*9*9*3^4*(3^2-1)*(3^4-1)*2;
true


#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> v:=[z,z^2,z^3,0*z,z^3,z,z,z^2,z^3,z^2,0*z,z,z^3,z^0,z^2,z^3,z,z^2];;
gap> result:=[];
gap> for k in [1..Size(genS)] do
> L:=Subgroup(GL(18,5),genS[k]);;
> H:=Normalizer(N,L);
> n:=Size(H);;
> m:=Size(v^H);
> result[k]:=[k,n,n/m];
> Display(result[k]);
> od;
[ 1, 155520, 1 ]
[ 2, 155520, 1 ]
[ 3, 746496, 1 ]
[ 4, 497664, 1 ]
[ 5, 155520, 1 ]
[ 6, 1492992, 1 ]
[ 7, 1492992, 1 ]
[ 8, 746496, 1 ]
[ 9, 497664, 1 ]
[ 10, 248832, 1 ]
[ 11, 248832, 1 ]
[ 12, 124416, 1 ]
[ 13, 746496, 1 ]
[ 14, 155520, 1 ]
[ 15, 7464960, 1 ]
[ 16, 746496, 1 ]
[ 17, 497664, 1 ]
[ 18, 497664, 1 ]
[ 19, 248832, 1 ]
[ 20, 248832, 1 ]
[ 21, 746496, 1 ]
[ 22, 373248, 1 ]
[ 23, 373248, 1 ]
[ 24, 373248, 1 ]
[ 25, 373248, 1 ]
[ 26, 4478976, 1 ]
[ 27, 497664, 1 ]
[ 28, 497664, 1 ]
[ 29, 746496, 1 ]
[ 30, 746496, 1 ]
[ 31, 746496, 1 ]
[ 32, 746496, 1 ]
[ 33, 373248, 1 ]
[ 34, 1244160, 1 ]
[ 35, 1492992, 1 ]
[ 36, 4478976, 1 ]
[ 37, 4478976, 1 ]
[ 38, 2239488, 1 ]
[ 39, 746496, 1 ]
[ 40, 1244160, 1 ]
[ 41, 4478976, 1 ]
[ 42, 1492992, 1 ]
[ 43, 4478976, 1 ]
[ 44, 4478976, 1 ]