$\mathrm{PSL}_3(4)$ inside the Monster group

Question

Which quasisimple groups with central quotient $G\cong\mathrm{PSL}_3(4)$ are isomorphic to subgroups of the Monster sporadic group? So far I know that $G$ itself is not and that $2\cdot G$, $2^2\cdot G$, and $6\cdot G$ are, which leaves $3\cdot G$, both $4\cdot G$s, $(2\times4)\cdot G$, both $12\cdot G$s, $(2\times6)\cdot G$, $4^2\cdot G$, $(4\times6)\cdot G$, and $(4\times12)\cdot G$.

Answer 1

The group $3.G$ centralizes an element of order three. If $3.G$ is a subgroup of the Monster then it is contained in a 3A centralizer (structure $3.Fi_{24}'$), a 3B centralizer (structure $3^{1+12}_+.2Suz$) or a 3C centralizer (structure $3 \times Th$). Clearly the case 3C cannot occur, and 3B is excluded by the fact that no class fusion between $3.G$ and the 3B normalizer $3^{1+12}_+.2Suz.2$ is possible --GAP's PossibleClassFusions can be used to show this, using the known character tables of the two groups. If $3.G$ is contained in the 3A centralizer then this embedding induces one of $G$ into some maximal subgroup of $Fi_{24}'$. Using the known character tables of these maximal subgroups in GAP's character table library, one shows that only $Fi_{23}$ admits a class fusion, but this subgroup lifts to $3 \times Fi_{23}$ in $3.Fi_{24}'$ and thus cannot lead to a subgroup of type $3.G$.

The other candidates $m.G$ contain at least one central involution. If $m.G$ is a subgroup of the Monster then it is contained in a 2A centralizer (structure $2.B$) or a 2B centralizer (structure $2^{1+24}_+.Co_1$). Again we use GAP's PossibleClassFusions to list all candidates for the class fusion, but here we prescribe the central involution of the 2A or 2B centralizer as an image of one central involution in $m.G$. It turns out that exactly one group $m.G$ cannot be excluded this way. Namely, these character-theoretical criteria leave the possibility that $4_1.G$ may occur as a subgroup of $2^{1+24}_+.Co_1$.