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We calculate all inequivalent irreducible representations of $\S_4$ byspecifying the matrices for adjacent transpositions and indicating how toobtain general permutations in $\S_4$ from these transpositions. We employstandard Young tableaux methods as found in Sagan's \emph{The Symmetric Group:Representations, Combinatorial Algorithms, and Symmetric Functions} (2001).

Young's Natural Representations of $\mathcal{S}_4$