Subgroups of $G(n,p)$ containing $SL(2,p)$ in an irreducible representation of degree $n$

In this paper we prove the following theorem.
Theorem. Suppose that $p>3n/2+1$ for $n<8$ and $p>2n-5$ for $n\geqslant8$, and $G$ is a subgroup of $GL(V_n)$ containing $\varphi_n(SL(2,p))$. Then one of the following assertions is true:
$1)$ $G\subset P^*\varphi_n(GL(2,p))$;
$2)$ $G\supset SL(n,p)$;
$3)$ $n$ is even and $Sp(n,p)\subset G\subset HSp(n,p)$;
$4)$ $n$ is odd and $\Omega(n,p)\subset G\subset P^*O(n,p)$;
$5)$ $n=7$ and $G=G_2(p)Z(G)$.
Here $P^*$ is the multiplicative group of the field $P$, $Sp(n,p)$ is the symplectic group, $HSp(n,p)$ is the group of symplectic similarities, $\Omega(n,p)$ is the derived group of the orthogonal group, $G_2(p)$ is the Chevalley group over $P$ associated with the Lie algebra of type $G_2$, and $Z(G)$ is the center of $G$.
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