  In this article, we construct two one-parameter families of properly embedded
minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is
the universal covering of the group of rigid motions of Euclidean plane endowed
with a left-invariant Riemannian metric. The first one can be seen as a family
of helicoids, while the second one is a family of catenoidal minimal surfaces.
The main tool that we use for the construction of these surfaces is a
Weierstrass-type representation introduced by Meeks, Mira, P\'erez and Ros for
minimal surfaces in Lie groups of dimension three. In the end, we study the
limit of the catenoidal minimal surfaces. As an application of this limit case,
we get a new proof of a half-space theorem for minimal surfaces in
$\widetilde{E(2)}$.
