On generating triples of involutions of large sporadic groups

In each finite simple sporadic group, excepting the Baby Monster group $B$, the Monster group $M$, the McLaughlin group $\mathit{McL}$ and Mathieu groups $M_{11}$, $M_{22}$, $M_{23}$, three generating involutions, two of which commute, are found.
If $G$ is one of the groups $M_{12}$, $M_{24}$, $\mathit{HS}$, $J_1$, $J_2$, $J_3$, then we give pairs of numbers $p$, $q$, $p\le q$, such that $p=|ik|$, $q=|jk|$ for some involutions $i$, $j$, $k$ with condition $|ij|=2$ generating the group $G$. The triples of involutions mentioned above are found with the use of the system of computer algebra GAP\@. Recall that any two involutions of the triple of involutions generating either $\mathit{McL}$, or $M_{11}$, or $M_{22}$, or $M_{23}$ do not commute.
This research was supported by the Russian Foundation for Basic Research, grant 02–01–00078.