  Modular Tensor Categories (MTC's) arise in the study of certain condensed
matter systems. There is an ongoing program to classify MTC's of low rank, up
to modular data. We present an overview of the methods to classify modular
tensor categories of low rank, applied to the specific case of a rank 6
category with Galois group $\langle(012)(345)\rangle$, and show that certain
symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We
show that all the rank 6 MTC's with this Galois group have modular data
conjugate to either the product of the semion category with $(A_1,
5)_{\frac{1}{2}}$ or a certain modular subcategory of
$\mathcal{C}(\mathfrak{so}_5, 9, e^{j\pi i/9})$ with gcd$(18, j) = 1$.
