Abstract
We construct a rank five residually connected and firm geometry Γ on which the Mathieu group M12 acts flag-transitively and residually weakly primitively (RWPRI). The group M12 is the group of automorphisms of Γ and Aut(M12) is the correlation group of Γ, in particular Γ is self-dual. The diagram of Γ is the following. Moreover Γ satisfies the conditions (IP)2 and (2T)1. As a corollary, we obtain that the (RWPRI+(IP)2)-rank of M12 is 5.
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