6. In $\triangle A B C$, the maximum value of $\sin A+\sin B+2 \sqrt{7} \sin C$ is $\qquad$ .
6. $\frac{27}{4}$.

Notice,
$$
\begin{array}{l}
\sin A+\sin B=2 \sin \frac{A+B}{2} \cdot \cos \frac{A-B}{2} \\
\leqslant 2 \sin \frac{A+B}{2}=2 \cos \frac{C}{2} .
\end{array}
$$

Let $\cos \frac{C}{2}=m$.
$$
\begin{array}{l}
\text { Then the original expression } \leqslant 2 \sqrt{7} \sin C+2 \cos \frac{C}{2} \\
=4 \sqrt{7} m \sqrt{1-m^{2}}+2 m \leqslant \frac{27}{4} \\
\Leftrightarrow 16 \sqrt{7} m \sqrt{1-m^{2}} \leqslant 27-8 m \\
\Leftrightarrow 1792 m^{2}\left(1-m^{2}\right) \leqslant 729+64 m^{2}-432 m \\
\Leftrightarrow(4 m-3)^{2}\left(112 m^{2}+168 m+81\right) \geqslant 0 .
\end{array}
$$

The above inequality clearly holds.
Therefore, the original expression $\leqslant \frac{27}{4}$.
Equality holds if and only if $\sin A=\sin B=\frac{3}{4}, \sin C=\frac{3 \sqrt{7}}{8}$,

in which case the original expression achieves its maximum value of $\frac{27}{4}$.