The generalized dihedral groups $Dih(\mathbb{Z}^n)$ as groups generated by time-varying automata

Let $\mathbb{Z}^n$ be a cubical lattice in the Euclidean space $\mathbb{R}^n$. The generalized dihedral group $Dih(\mathbb{Z}^n)$ is a topologically discrete group of isometries of $\mathbb{Z}^n$ generated by translations and reflections in all points from $\mathbb{Z}^n$. We study this group as a group generated by a $(2n+2)$-state time-varying automaton over the changing alphabet. The corresponding action on the set of words is described.