In this article, we investigate the representations of the Drinfeld doubles $D(R_{ mn}(q))$ of the Radford Hopf algebras $R_{ mn}(q)$ over an algebraically closed field $\Bbbk$, where $m>1$ and $n>1$ are integers and $q\in\Bbbk$ is a root of unity of order $n$. Under the assumption ${ \rm char}(\Bbbk)\nmid mn$, all the finite dimensional indecomposable modules over $D(R_{ mn}(q))$ are displayed and classified up to isomorphism. The Auslander-Reiten sequences in the category of finite dimensional $D(R_{ mn}(q))$-modules are also all displayed. It is shown that $D(R_{ mn}(q))$ is of tame representation type.
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