Exotic Fusion Systems Related to Sporadic Simple Groups (Version With Code)

We describe several exotic fusion systems related to the Sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow $3$-subgroups of the Conway group $\mathrm{Co}_1$ and the Thompson group $\mathrm{F}_3$, and a Sylow $5$-subgroup of the Monster $\mathrm{M}$, as well as a particular maximal subgroup of the latter two $p$-groups. This work is supported by computations in MAGMA.