In this work, we have modified a scenario, originally proposed by Grimus and Lavoura, in order to obtain maximal values for atmospheric mixing angle and $CP$ violating Dirac phase of the lepton sector. To achieve this, we have employed $CP$ and some discrete symmetries in a type II seesaw model. In order to make predictions about neutrino mass ordering and the smallness of the reactor angle, we have obtained some conditions on the elements of the neutrino mass matrix of our model. Finally, within the framework of our model, we have studied quark masses and mixing pattern. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.