Let $\sin{A} =4k$, $\sin{B}=5k$, and $\sin{C}=7k$, respectively, where $k$ is a constant. By the Law of Sine, we have $$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}\implies \frac{a}{4k}=\frac{b}{5k}=\frac{c}{7k}$$ Set the ratio be $l$ yields:$$a = 4kl, b=5kl, c=7kl$$By the law of cosine, we have$$\cos{C} = \frac{a^2+b^2-c^2}{2ab}=\frac{16k^2l^2 + 25k^2l^2 -49k^2l^2}{2\times 4kl \times 5kl}=\boxed{-\frac{1}{5}}$$ It follows that (note $0 < C < \pi\implies \sin{C} > 0$) $$\sin{C}=\sqrt{1-\cos^2{C}}=\sqrt{1-\Big(-\frac{1}{5}\Big)^2}=\boxed{\frac{2}{5}\sqrt{6}}$$