The existence of triple factorizations for sporadic groups of rank 3

A finite group $G$ with proper subgroups $A$ and $B$ has triple factorization $G = ABA$ if every element $g$ of $G$ can be represented as $g = aba'$, where $a$ and $a'$ are from $A$ and $b$ is from $B$. Such a triple factorization may be sometimes degenerate to $AB$-factorization.
The task of finding triple factorizations for a group is fundamental and can be used for understanding the group structure. For instance, every simple finite group of Lie type has a natural factorization of such a type. Besides, the triple factorization is widely used in the study of graphs, geometries and varieties.
The goal of this article is to find triple factorizations for sporadic groups of rank $3$. We have proved the existence theorem of $ABA$-factorization for sporadic simple groups $McL$ and $Fi_{22}$. There exist two rank $3$ permutation representations of $Fi_{22}$. We have proved that $ABA$-factorizations exist in both cases.