A gadolinium-scandium-aluminum garnet single crystal (Gd3Sc2Al3O12, GSAG) co-doped with 2%Dy3+ and 1% Tb3+ (atomic number fraction) was grown by Czochralski method. The morphology of etched defect on the crystal surface of (111) plane was studied and the mechanism for the formation of the defects was discussed. Besides, the Vickers hardness and Mohs hardness of the (111) plane were also characterized and calculated. On the condition of 0.2 kgf loading and 10 s keeping time, the Vickers hardness for the (111) plane is 1 267 kg/mm2, corresponding to the Mohs hardness of 7.3. The results have reference value for revealing the origin of defects in mixed garnet crystals and exploring the growth technics for high quality crystals.