We present here a detailed investigation into the magnetic ordering of full Heusler alloy Ga$_2$MnCo using dc, ac magnetization measurements, neutron diffraction and neutron depolarization experiments. Crystal structure at room temperature was first confirmed to be L2$_1$ using the highly intense synchrotron X-ray diffraction (XRD) technique. Temperature dependent magnetization reveals that Ga$_2$MnCo enters a ferromagnetic (FM) state at $T_C = $154 K, characterized by a sharp increase in magnetization and a plateau-like region hereafter. As the temperature is decreased further, a sharp drop in magnetization is observed at $T_f$ = 50 K, hinting towards an antiferromagnetic (AFM) phase change. Neutron diffraction (ND) recorded over the range of temperature from 6 to 300 K, provides combined information regarding crystal as well as magnetic structure. Accordingly, an increase in the intensity of the ND pattern is seen at 150 K, signaling onset of long range FM order. However, there is no sign of appearance of superlattice reflections corresponding to the AFM phase, in the patterns recorded below 50 K. An unusual discontinuity in the unit cell volume is seen around $T_f$ indicating a coupling of this second transition with the contraction of the lattice. Attempts to unravel this interesting magnetic behaviour using ac susceptibility measurements lead to the existence of glassy magnetism below $T_f$. Systematic analysis of the susceptibility results along with neutron depolarization measurement, identifies the low temperature phase as a reentrant cluster glass.