  We introduce extensions of the multidimensional Heisenberg group
$\mathbb{H}^n$ by two-parameter groups of dilations, and then classify the
extended groups up to isomorphism, by employing Lie algebra techniques. We show
that the groups are isomorphic to subgroups of the symplectic group
$\textit{Sp(}n+1,\mathbb{R})$ as well as subgroups of the affine group
$\textit{Aff}(n+1,\mathbb{R})$. Thus, they possess both, a metaplectic and a
wavelet representation. Moreover, the metaplectic representation splits into a
sum of two subrepresentations which both are equivalent to the same
subrepresentation of the wavelet representation.
