  Harnessing non-abelian statistics of anyons to perform quantum computational
tasks is getting closer to reality. While the existence of universal anyons by
braiding alone such as the Fibonacci anyon is theoretically a possibility,
accessible anyons with current technology all belong to a class that is called
weakly integral---anyons whose squared quantum dimensions are integers. We
analyze the computational power of the first non-abelian anyon system with only
integral quantum dimensions---$D(S_3)$, the quantum double of $S_3$. Since all
anyons in $D(S_3)$ have finite images of braid group representations, they
cannot be universal for quantum computation by braiding alone. Based on our
knowledge of the images of the braid group representations, we set up three
qutrit computational models. Supplementing braidings with some measurements and
ancillary states, we find a universal gate set for each model.
