  Representation theory of an infinite dimensional Galilean conformal algebra
introduced by Martelli and Tachikawa is developed. We focus on the algebra
defined in (2+1) dimensional spacetime and consider central extension. It is
then shown that the Verma modules are irreducible for non-vanishing highest
weights. This is done by explicit computation of Kac determinant. We also
present coadjoint representations of the Galilean conformal algebra and its Lie
group. As an application of them, a coadjoint orbit of the Galilean conformal
group is given in a simple case.
