Solution
= Solution
Put
$$
A=\begin{pmatrix}0&\varepsilon\\\varepsilon&0\end{pmatrix},
\qquad
B=\begin{pmatrix}\omega&0\\0&\omega^2\end{pmatrix}.
$$
Then $A^2=\varepsilon^2I$, so $A^{2n}=I$ because $\varepsilon^{2n}=1$, and $B^3=I$. Moreover, $A$ exchanges the two eigenspaces of $B$, giving
$$
A^{-1}BA=B^{-1}.
$$
The matrices therefore satisfy every defining relation of $G_{6n}$, and the universal property of a group presentation gives the required <complex representation>.