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In this paper, by considering a supercharge $\\mathcal{Q}$ which is cubic in terms of the letters, we construct the Hamiltonian of the largest Spin Matrix theory of $\\mathcal{N}=4$ SYM, called the PSU$(1,2|3)$ Spin Matrix theory, as $H = \\{\\mathcal{Q}, \\mathcal{Q}^\\dagger \\}$. 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