  A detailed study of the magnetic and magnetocaloric properties of a garnet
compound Mn$_{3}$Cr$_{2}$Ge$_{3}$O$_{12}$ is carried out using x-ray
diffraction, magnetization, heat capacity, and neutron diffraction measurements
as well as \textit{ab initio} band-structure calculations. This compound
manifests two successive magnetic transitions at $T_{\rm N1} \simeq 4.5$ K and
$T_{\rm N2} \simeq 2.7$ K. Neutron powder diffraction experiments reveal that
these two transitions correspond to the collinear and non-collinear
antiferromagnetic ordering of the nonfrustrated Cr$^{3+}$ and frustrated
Mn$^{2+}$ sublattices, respectively. The interactions within each of the Cr and
Mn sublattices are antiferromagnetic, while the inter-sublattice 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. Mn$_{3}$Cr$_{2}$Ge$_{3}$O$_{12}$ shows a large
magnetocaloric effect with a maximum value of isothermal entropy change $\Delta
S_{\rm m} \simeq -23$ J/kg-K and adiabatic temperature change $\Delta T_{\rm
ad} \simeq 9$ 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.
