= Solution
For the order-18 group, the index-two subgroup $P$ is normal and $G=\langle P,t\rangle$. Restrict a complex <irreducible representation> $V$ to the finite abelian group $P$. Simultaneous diagonalization decomposes it into character spaces $V_\chi$. The inversion relation makes $t$ exchange $V_\chi$ and $V_{\chi^{-1}}$. The sum of spaces belonging to any such character orbit is $G$-invariant, so irreducibility leaves only one orbit.
If the character is trivial, all of $P$ is in the representation's kernel. Otherwise $\chi$ has order three and $\chi\ne\chi^{-1}$. The only spaces are $V_\chi$ and $V_{\chi^{-1}}$, and both have the same kernel in $P$. A nontrivial character of $P\cong C_3\times C_3$ maps onto $C_3$, so that kernel has order three. It acts trivially on all of $V$. Therefore \b[the group has no faithful irreducible complex representation]. In fact choosing nonzero $v\in V_\chi$ makes $\operatorname{span}(v,tv)$ invariant, so these nontrivial-character irreducibles have degree two.
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