  In this paper, we consider the optimization problem for the first Dirichlet
eigenvalue $\lambda_1(\Omega)$ of the $p$-Laplacian $\Delta_p$, $1< p< \infty$,
over a family of doubly connected planar domains $\Omega= B \setminus
\overline{P}$, where $B$ is an open disk and $P\subsetneq B$ is a domain which
is invariant under the action of a dihedral group $\mathbb{D}_n$ for some $n
\geq 2,\;n\in \mathbb{N}$. We study the behaviour of $\lambda_1$ with respect
to the rotations of $P$ about its center. We prove that the extremal
configurations correspond to the cases where $\Omega$ is symmetric with respect
to the line containing both the centers. Among these optimizing domains, the
OFF configurations correspond to the minimizing ones while the ON
configurations correspond to the maximizing ones. Furthermore, we obtain
symmetry (periodicity) and monotonicity properties of $\lambda_1$ with respect
to these rotations. In particular, we prove that the conjecture formulated in
[14] for $n$ odd and $p=2$ holds true. As a consequence of our monotonicity
results, we show that if the nodal set of a second eigenfunction of the
$p$-Laplacian possesses a dihedral symmetry of the same order as that of $P$,
then it can not enclose $P$.
