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A general construction procedure of the allowed states of the multi-cluster system is presented with the use of the concept of the "coefficient of fractional parentage". We explain in detail in the case of three-cluster system that the introduction of the Elliott SU3 group greatly simplifies the construction process, thereby making the practical calculation significantly rapid. As an important application of the allowed states, we discuss the calculation of the spectroscopic amplitude ofdoi:10.1143/ptp.58.204 fatcat:uu7ewxhj6rcgtcf7j5v4n5mifa