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In this work we study and exactly solve the Dirac oscillator with three different topological defects, namely the cosmic string spacetime ($\Lambda_\mp$), the magnetic cosmic string spacetime ($\Theta_\mp$) and the cosmic dislocation spacetime ($\Pi_\mp$). Moreover, we show that the radial part of this problem possess an $SU(1,1)$ symmetry. From this, we obtain the wave functions and their respective energy spectrum by means of the Schr\"odinger factorization. Also, we compute the radial coherent states of the eigenfunctions of each problem and their time evolution., Comment: 13 pages