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The structure of a Dirichlet group of characters of modulus $q$ refers to its isomorphism type as a finite abelian group, which can be uniquely expressed as a product of cyclic groups $C_{n_1}\times C_{n_2}\cdots\times C_{n_r}$ with $n_1|n_2|\cdots |n_r$. This is the same as the structure of the group $(\Z/q\Z)^\times$.