  The Maxwell group in 2+1 dimensions is given by a particular extension of a
semi-direct product. This mathematical structure provides a sound framework to
study different generalizations of the Maxwell symmetry in three space-time
dimensions. By giving a general definition of extended semi-direct products, we
construct infinite-dimensional enhancements of the Maxwell group that enlarge
the ${\rm ISL}(2,\mathbb{R})$ Kac-Moody group and the ${\rm BMS}_3$ group by
including non-commutative supertranslations. The coadjoint representation in
each case is defined, and the corresponding geometric actions on coadjoint
orbits are presented. These actions lead to novel Wess-Zumino terms that
naturally realize the aforementioned infinite-dimensional symmetries. We
briefly elaborate on potential applications in the contexts of
three-dimensional gravity, higher-spin symmetries, and quantum Hall systems.
