On finite groups with an irreducible character large degree

Let $G$ be a finite nontrivial group with an irreducible complex character $\chi$ of degree $d=\chi(1)$. It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of $G$ is equal to the order of $G$. N. Snyder proved that if $|G|=d(d+e)$, then the order of $G$ is bounded in terms of $e$, provided $e>1$. Y. Berkovich proved that in the case $e=1$ the group $G$ is Frobenius with the complement of order $d$. We study a finite nontrivial group $G$ with an irreducible complex character $\Theta$ such that $|G|\leq2\Theta(1)^2$ and $\Theta(1)=pq$, where $p$ and $q$ are different primes. In this case we prove that $G$ is solvable groups with abelian normal subgroup $K$ of index $pq$. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime $p$ and of order less than $2p^4 $ is isomorphic to $L_2(q), L_3(q), U_3(q), Sz(8), A_7, M_{11}$ or $J_1$.