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

Triply Extended Group of Translations of $\mathbb{R}^{4}$ as Defining Group of NCQM: relation to various gauges