  We carry out a detailed analysis of the nonminimal supersymmetric standard
model with lepton number violation. The model contains a unique trilinear
lepton number violating term in the superpotential which can give rise to
neutrino masses at the tree level. We search for the gauged discrete symmetries
realized by cyclic groups which preserve the structure of the associated
trilinear superpotential of this model, and which satisfy the constraints of
the anomaly cancellation. The implications of this trilinear lepton number
violating term in the superpotential and the associated soft supersymmetry
breaking term on the phenomenology of the light neutrino masses and mixing is
studied in detail. We evaluate the tree and loop level contributions to the
neutrino mass matrix in this model. We search for possible suppression
mechanism which could explain large hierarchies and maximal mixing angles.
