Abstract. With linearly independent elements extracted from 3×3 unitary matrix representations Rep(S₃) of the permutation group S₃, a non-semisimple algebra and its associated Lie group are constructed for the purpose of describing flavor physics. While the group is isomorphic to the SU(2)×U(1) group, the algebra has a unique, elaborate structure. A flavor-mixing matrix (FMM) that is completely different from that of the conventional theory is expressed in terms of four group parameters of the Lie group in an essentially unique way. Adjusting the values of the group parameters of the FMM for quark sector allows us to account for experimental results to high precision. Unique elaborate structure of the algebra leads also to sum rules of elements of the FMM that are so simple and clean-cut to be verifiable by analyzing accumulated experimental data.

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