4. In the Cartesian coordinate system $x O y$, let $F_{1}, F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, respectively. $P$ is a point on the right branch of the hyperbola, $M$ is the midpoint of $P F_{2}$, and $O M \perp P F_{2}, 3 P F_{1}=4 P F_{2}$. Then the eccentricity of the hyperbola is $\qquad$.
According to the problem, $\left|P F_{1}\right|=8 a, \mid P F_{2}=6 a$, then $\left|F_{1} F_{2}\right|=10 a=2 c \Rightarrow e=\frac{c}{a}=5$.

