  The Lie algebra of the symmetry group of the $(n+1)$-dimensional
ge\-ne\-ra\-li\-zation of the dispersionless Kadomtsev--Petviashvili (dKP)
equation is obtained and identified as a semi-direct sum of a finite
dimensional simple Lie algebra and an infinite dimensional nilpotent
subalgebra. Group transformation properties of solutions under the subalgebra
$\Sl(2,\mathbb{R})$ are presented. Known explicit analytic solutions in the
literature are shown to be actually group-invariant solutions corresponding to
certain specific infinitesimal generators of the symmetry group.
