Representations of the modular group arising from Drinfeld doubles of finite abelian groups

We show that the image of the representation of the modular group SL(2,Z) arising from the representation category Rep(D(G)) of the Drinfeld double D(G) of a finite abelian group G of exponent n is isomorphic to the special linear group SL(2,Z/nZ), where Z/nZ denotes the ring of integers modulo n. As a consequence, we establish that the kernel of the representation in question is the principal congruence subgroup of level n.