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Leopardi introduced the notion of a Kronecker quotient in [Paul Leopardi. A generalized FFT for Clifford algebras. Bulletin of the Belgian Mathematical Society, 11:663–688, 2005.]. This article considers the basic properties that a Kronecker quotient should satisfy and additional properties which may be satisfied. A class of Kronecker quotients for which these properties have a natural description is completely characterized. Two examples of types of Kronecker quotients are described.