  Rare earth garnets are an exciting playground for studying the exotic
magnetic properties of the frustrated hyperkagome lattice. Here we present a
comprehensive study of the single ion and collective magnetic properties of the
garnet Er$_3$Ga$_5$O$_{12}$. Using inelastic neutron scattering, we find a
crystal field ground state doublet for Er$^{3+}$ with strong Ising anisotropy
along local [100] axes. Magnetic susceptibility and heat capacity measurements
provide evidence for long-range magnetic ordering with $T_N$~$=$~0.8~K, and no
evidence for residual entropy is found when cooling through the ordering
transition. Neutron powder diffraction reveals that the ground state spin
configuration corresponds to the six-sublattice, Ising antiferromagnetic state
($\Gamma_3$) common to many of the rare earth garnets. However, we also found
that $\mu$SR appears to be insensitive to the ordering transition in this
material, in which a low-temperature relaxation plateau was observed with no
evidence of spontaneous muon precession. The combined muon and neutron results
may be indicative of a dynamical ground state with a relatively long
correlation time. Despite this potential complication, our work indicates that
Er$_3$Ga$_5$O$_{12}$ is an excellent model system for studying the complex
metamagnetism expected for a multi-axis antiferromagnet.
