A classification of class two and class three nilpotent table algebras
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Table algebras are generalizations of adjacency algebras, and of the character ring of a finite group. Extensions of groups by groups have been well studied, and Hirasaka and Bang have generalized this to the study of extensions of association schemes by association schemes. In this dissertation, we study central extensions of table algebras by table algebras, in the case where the extension is either class two nilpotent (which means it is an extension of a group algebra by a group algebra), or class three nilpotent (which means it is an extension of a class two nilpotent table algebra by a group algebra) with order p3 for an arbitrary prime p. We classify these algebras up to exact isomorphism. In the class two case, we determine exactly when the algebra is the adjacency algebra of an association scheme, and in the class three case, we determine which sets of the parameters of our classication determine isomorphic algebras.