A new existence proof of in a computational approach

The purpose of this article is to give a new, explicit and elementary construction of the (Formula presented.) in (Formula presented.) using the notion of M-sets introduced by the second author, and properties of the generalized quadrangle (Formula presented.) of type (Formula presented.) This construction is elementary and explicit as the transpositions generating (Formula presented.) in (Formula presented.) have been explicitly constructed and implemented in GAP, which might be very helpful and well suited to people who do computations with this group. It is remarkable to mention that this work supports the work of Cuypers et al., where the existence of (Formula presented.) has been assumed then the embedding of the sporadic simple group (Formula presented.) in the group (Formula presented.) has been given. In fact our approach for the construction is completely different from Fischer’s construction.