Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors such that
\[\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 6 \\ -7 \\ 3 \end{pmatrix}, \quad \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix}, \quad \mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix}.\]Compute $\mathbf{c} \times (3 \mathbf{a} - 2 \mathbf{b}).$
Expanding, we get
\begin{align*}
\mathbf{c} \times (3 \mathbf{a} - 2 \mathbf{b}) &= 3 \mathbf{c} \times \mathbf{a} - 2 \mathbf{c} \times \mathbf{b} \\
&= -3 \mathbf{a} \times \mathbf{c} + 2 \mathbf{b} \times \mathbf{c} \\
&= -3 \begin{pmatrix} 4 \\ 7 \\ 2 \end{pmatrix} + 2 \begin{pmatrix} 1 \\ -7 \\ 18 \end{pmatrix} \\
&= \boxed{\begin{pmatrix} -10 \\ -35 \\ 30 \end{pmatrix}}.
\end{align*}