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In this paper, we investigate the low temperature magnetic properties of the rare-earth garnet compound Nd$_3$Ga$_5$O$_{12}$ in detail by means of magnetization, specific heat and magnetocaloric effect measurements. The magnetic thermal properties along with the crystal field calculations reveal that the Nd$^{3+}$ ions form into a frustrated hyper-kagome lattice with connected triangles have an Ising-like ground state with the easy axis along the local [100], [010] and [001] directions. Instead of a quantum spin liquid ground state, an antiferromagnetically ordered state is found below $T_{\mathrm{N}}=0.52~\rm K$. With applying field in the [111] direction, the antiferromagnetic order is suppressed at the critical field of $B_{\mathrm{c}}=0.75~\rm T$, and enhancement of the critical fluctuations with linear crossover behaviors is observed near the critical point.