In this paper, the hyperkagome lattice of Gd spins in a garnet compound, Gd3CrGa4O12, is studied using bulk measurements and density functional computations, and the observation of large magnetocaloric effect corresponding to an entropy change, ΔSm = 45 J kg-1K-1 (≈ 45 J mol-1K-1) at 2 K, 8 T is reported. Though the compound defies long-range magnetic order down to 0.4 K, a broad feature below 10 K is observed in the specific heat with two low temperature anomalies at T* ≈ 0.7 K and TS ≈ 2.45 K. The anomaly at T* is reminiscent of one in Gd3Ga5O12, where it is related to the development of a complex magnetic phase, whereas the TS-peak is accounted for by a multilevel Schottky-like model. The spin-lattice relaxation times studied by nuclear magnetic resonance experiments show that the relaxation is dominated by the magnetic fluctuations in Cr which has a longer relaxation time compared to that of the garnet, Lu3CrGa4O12 containing a nonmagnetic rare earth. Our first-principles density functional theory calculations agree well with the experimental results and support short-range magnetic order in the Gd-sublattice and antiferromagnetism in the Cr-sublattice. The importance of spin fluctuations and short-range order in the rare earth and transition metal lattices in garnets resulting in large magnetocaloric effect is brought out through this work.