  We explore the group theoretical underpinning of noncommutative quantum
mechanics for a system moving on the two-dimensional plane. We show that the
pertinent groups for the system are the two-fold central extension of the
Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of
the group of translations of $\mathbb R^4$. This latter group is just the
standard Weyl-Heisenberg group of standard quantum mechanics with an additional
central extension. We also look at a further extension of this group and
discuss its significance to noncommutative quantum mechanics. We build unitary
irreducible representations of these various groups and construct the
associated families of coherent states. A coherent state quantization of the
underlying phase space is then carried out, which is shown to lead to exactly
the same commutation relations as usually postulated for this model of
noncommutative quantum mechanics.
