Finite Group Reduction of the DR/DZ Hierarchies

We show that the finite group reduction of the Dubrovin-Zhang/Double Ramification hierarchy associated to a semisimple CohFT preserves its bihamiltonian structure and its tau structure. By applying this result to the orbifold Gromov-Witten theory with the target ${\mathbb{P}}_{2,2,2,2}^1$, we see that there are two different and interesting integrable hierarchies that can be associated to the Frobenius manifold structure on the Hurwitz space $M_{1;1}$. One of them is equivalent to the genus $1$ topological recursion.