[en] The formation of hexagonal current pattern in a dc-driven thin planar gas discharge gap with a semiconductor cathode is studied on the basis of a 3D drift-diffusion discharge model. The model includes continuity equations for electron and ion densities, the electrostatic equation for electric field and heat conduction equation for gas temperature. Solving these equations by the time-relaxation method, we investigate the genesis of ordered current pattern and its subsequent rearrangement into the hexagonal pattern, starting from initial homogeneous discharge state without any initial perturbations. For the case of the dc discharge with a semiconductor cathode and cryogenic conditions, the 3D calculation has been performed for the first time. The resulting current pattern is in agreement with experiment. The results of the simulation confirm that the discharge instability and the patterns formation are due to an ionization-overheating (thermal) mechanism, as was proposed before in [Raizer Yu P and Mokrov M S 2010 J. Phys. D: Appl. Phys. 43 225204] on the basis of approximate theory. (paper)