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We study the quantum double of a finite abelian group G twisted by a 3-cocycle and give a sufficient condition when such a twisted quantum double will be gauge equivalent to a ordinary quantum double of a finite group. Moreover, we will determine when a twisted quantum double of a cyclic group is genuine. As an application, we contribute to the classification of coradically graded finite-dimensional pointed coquasi-Hopf algebras over abelian groups. As a byproduct, we show that the Nichols algebras B (M₁ M₂ M₃) are infinite-dimensional where M₁, M₂, M₃ are three different simple Yetter-Drinfeld modules of D₈.
Li et al. (Sun,) studied this question.