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The paper studies the possibility of giving a general multiplicity of the fractional Fourier transform (FRFT) with the intention of combining existing finite versions of the FRFT. We introduce a new class of FRFT that includes the conventional fractional Fourier transforms (CFRFTs) and the weighted-type fractional Fourier transforms (WFRFTs) as special cases. The class is structurally well organized because these new FRFTs, which are called general multifractional Fourier transform (GMFRFTs), are related with one another by the Generalized Permutation Matrix Group (GPMG), and their kernels are related with that of CFRFTs as the finite combination by the recursion of matrix. In addition, we have computer simulations of some GMFRFTs on a rectangular function as a simple application of GMFRFTs to signal processing.