#We find all primitive subgroups of $GL(4,67)$ with parameters $e=4$, $a=1$. Since in this case $b=2$ does not divide $a=1$, Lemma 5.3 implies that there exists 
#two nonconjugate subgroups $F_1$ and $F_2$ such that $N_{GL(4,67)}(F_1)/F_1\simeq O^+(4,2)$, while $N_{GL(4,67)}(F_2)/F_2\simeq O^-(4,2)$. This log-file contains
#calculations for $F_2$.
#In the notations of Theorem 5.1, if $H$ is a primitive solvable subgroup, then it possesses a series

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

#$U$ is cyclic of order $66$, $F$ is of type $2^{1+4}$ is such that $N_{GL(4,67)}(F)/F\simeq O^-(4,2)$, and $G=A$. First we take the list of 
#generators of maximal solvable primitive subgroup of $GL(4,67)$, containing all maximal primitive subgroups of $GL(4,67)$ from the file   generators-GL(4,67)-.txt.
#In view of Lemma 5.4, there are two maximal solvable subgroups of $O^-(4,2)$ with 2-radical of order at most 2: Frobenius group $5^4$ and 
#$O^+(2,2)\times O^-(2,2)\simeq S_2\times S_3$. The second group stabilizes nontrivial isotropic subspace, so we need  to consider only the full preimage of 
#the first group and gens contains generators of exactly one group.

#Now we run the following loop, where gens is the list of generators from generators-GL(4,67)-.txt. 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(67);; v:=[z,z^13,z^23,z^29];;
gap> result:=[];;
gap> H:=Subgroup(GL(4,67),j);; 
gap> n:=Size(H);; m:=Size(v^H);; 
gap> Display([n,n/m]);
[ 21120, 1 ]