= Solution
Every element has a unique form $a^s b^t$. Since $a^2$ is central, the conjugacy classes, indexed by $0\leq r<n$, are
$$
C_r=\{a^{2r}\},\qquad
D_r=\{a^{2r}b,a^{2r}b^2\},\qquad
E_r=\{a^{2r+1},a^{2r+1}b,a^{2r+1}b^2\}.
$$
These have sizes one, two, and three and account for all $6n$ elements.
Take $\rho_k=\rho_{\xi^k}$ for $0\leq k<n$. On an even power of $a$, its representing matrix is scalar; on an odd power it is off-diagonal. Since $\omega+\omega^2=-1$, the full <Character table of G6n> is
$$
\boxed{
\begin{array}{c|ccc}
&C_r&D_r&E_r\\ \hline
\lambda_j&\xi^{2jr}&\xi^{2jr}&\xi^{j(2r+1)}\\
\chi_{\rho_k}&2\xi^{2kr}&-\xi^{2kr}&0
\end{array}
}
$$
for $0\leq j<2n$, $0\leq k<n$, and $0\leq r<n$.
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