The construction of a $E_7$-like quantum subgroup of $SU(3)$

In this short note we construct an embedding of the planar algebra for $\overline{\operatorname{Rep}(U_q(sl_3))}$ at $q = e^{2\pi i \frac{1}{24}}$ into the graph planar algebra of di Francesco and Zuber's candidate graph $\mathcal{E}_4^{12}$. Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over $\overline{\operatorname{Rep}(U_q(sl_3))}$ whose graph for action by the vector representation is $\mathcal{E}_4^{12}$. This fills a small gap in the literature on the construction of $\overline{\operatorname{Rep}(U_q(sl_3))}$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.