A detailed study of the magnetic and magnetocaloric properties of the garnet compound ${\mathrm{Mn}}_3{\mathrm{Cr}}_2{\mathrm{Ge}}_3{\mathrm{O}}_{12}$ is carried out using x-ray diffraction, magnetization, heat capacity, and neutron diffraction measurements as well as ab initio band-structure calculations. This compound manifests two successive magnetic transitions at $T_{\mathrm{N}1}\simeq 4.5\mathrm{K}$ and $T_{\mathrm{N}2}\simeq 2.7\mathrm{K}$. Neutron powder diffraction experiments reveal that these two transitions correspond to the collinear and noncollinear antiferromagnetic ordering of the nonfrustrated ${\mathrm{Cr}}^{3+}$ and frustrated ${\mathrm{Mn}}^{2+}$ sublattices, respectively. The interactions within each of the Cr and Mn sublattices are antiferromagnetic, while the intersublattice interactions are ferromagnetic. The H-T phase diagram is quite complex and displays multiple phases under magnetic field, which can be attributed to the frustrated nature of the spin lattice. ${\mathrm{Mn}}_3{\mathrm{Cr}}_2{\mathrm{Ge}}_3{\mathrm{O}}_{12}$ shows a large magnetocaloric effect with a maximum value of isothermal entropy change $\mathrm{\Delta }{S}_{\mathrm{m}}\simeq -23$ J/kg K and adiabatic temperature change $\mathrm{\Delta }{T}_{\mathrm{ad}}\simeq 9\mathrm{K}$ for a field change of 7 T. Further, a large value of the relative cooling power ($RCP\simeq 360$ J/kg) demonstrates the promise of using this compound in magnetic refrigeration.