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The planar Hall effect (PHE), wherein a rotating magnetic field in the plane of a sample induces oscillating transverse voltage, has recently garnered attention in a wide range of topological metals and insulators. The observed twofold oscillations in ρ_{yx} as the magnetic field completes one rotation are the result of chiral, orbital, and/or spin effects. The antiperovskites A_{3}BO (A=Ca, Sr, Ba; B=Sn, Pb) are topological crystalline insulators whose low-energy excitations are described by a generalized Dirac equation for fermions with total angular momentum J=3/2. We report unusual sixfold oscillations in the PHE of Sr_{3}SnO, which persisted nearly up to room temperature. Multiple harmonics (twofold, fourfold, and sixfold), which exhibited distinct field and temperature dependencies, were detected in ρ_{xx} and ρ_{yx}. These observations are more diverse than those in other Dirac and Weyl semimetals and point to a richer interplay of microscopic processes underlying the PHE in the antiperovskites.