?
On the coverings of Euclidean manifolds $\mathbb{G}_3$ and $\mathbb{G}_5$
There are only 10 Euclidean forms, that is flat closed three
dimensional manifolds: six are orientable and four are
non-orientable. The aim of this paper is to describe all types of
$n$-fold coverings over the orientable Euclidean manifolds
$\mathcal{G}_{3}$ and $\mathcal{G}_{5}$, and calculate the numbers
of non-equivalent coverings of each type. The manifolds
$\mathcal{G}_{3}$ and $\mathcal{G}_{5}$ are uniquely determined
among other forms by their homology groups
$H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})=
\ZZ$.
We classify subgroups in the fundamental groups
$\pi_1(\mathcal{G}_{3})$ and $\pi_1(\mathcal{G}_{5})$ up to
isomorphism. Given index $n$, we calculate the numbers of
subgroups and the numbers of conjugacy classes of subgroups for each
isomorphism type and provide the Dirichlet generating functions for
the above sequences.