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The compensation behaviors and phase transitions of a new 3D fullerene-like polymer are explored by using Monte Carlo method. The ground-state phase diagrams and magnetization profiles are given, and the conditions for the occurrence of the compensation temperature Tcomp are obtained. As classified in Néel theory, N-type and P-type magnetization curves are found. To explore the changes in the transition temperature TC and the compensation temperature Tcomp induced by physical parameters, we also present the phase diagrams. Furthermore, the critical exponents of the system are calculated, and a good agreement can be achieved by comparing our results with others'. Finally, the triple hysteresis loops are discovered. [ABSTRACT FROM AUTHOR]