  Let H be the 15- dimensional connected semisimple Lie group with its Iwasawa
decomposition of H. Let G be the group of the semi direct product of H and the
four dimensional real vector group . The goal of this paper is to define the
Fourier transform in order to obtain the Plancherel theorem on H and so on G.
Since its symplectic group of dimension 10 is a subgroup of H, then it will be
easy to get the Plancherel theorem on the symplectic group and so on its
inhomogeneous group. To this end, we obtain some interesting results on its
nilpotent symplectic group
