In this study, we modify a scenario, originally proposed by Grimus and Lavoura, in order to obtain maximal values for the atmospheric mixing angle and , violating the Dirac phase of the lepton sector. To achieve this, we employ and some discrete symmetries in a type II seesaw model. To make predictions about the neutrino mass ordering and smallness of the reactor angle, we establish some conditions on the elements of the neutrino mass matrix of our model. Finally, we study the quark masses and mixing pattern within the framework of our model. © 2022 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.