  In this article, we determine quantum dimensions and fusion rules for the
orbifold code VOA $ V^\tau_{L_{C \times D}}$. As an application, we also
construct certain $3$-local subgroups inside the automorphism group of the VOA
$V^\sharp$, where $V^\sharp$ is a holomorphic VOA obtained by the
$\mathbb{Z}_3$-orbifold construction on the Leech lattice VOA.
