Fusion systems over a Sylow $p$-subgroup of $\mathrm{G}_2(p)$

Abstract

For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.