  The role of the triply extended group of translations of $\mathbb{R}^{4}$, as
the defining group of two dimensional noncommutative quantum mechanics (NCQM),
has been studied in \cite{ncqmjmp}. In this paper, we revisit the coadjoint
orbit structure and various irreducible representations of the group associated
with them. The two irreducible representations corresponding to the Landau and
symmetric gauges are found to arise from the underlying defining group. The
group structure of the transformations, preserving the commutation relations of
NCQM, has been studied along with specific examples. Finally, the relationship
of a certain family of UIRs of the underlying defining group with a family of
deformed complex Hermite polynomials has been explored .
