Our calculations show that the III-VI GaSe semiconductor compound exhibits a slight preference to crystallize at zero pressure in structure rather than in structure; and under appropriate pressures, phase transitions occur toward NaCl (B1) cubic phase. The elastic constants are calculated for all these structures and compared with available previous results. The stability of GaSe in the rock salt (B1), CsCl (B2), Zinc blende (B3), Wurtzite (B4), and structures is then determined, followed by an electronic structure investigation.