Construction of Fischer ’ S Sporadic

In this article we construct an irreducible simple subgroup G = 〈q, y, t, w〉 of GL8671(13) from an irreducible subgroup T of GL11(2) isomorphic to Mathieu’s simple group M24 by means of Algorithm 2.5 of [13]. We also use the first author’s similar construction of Fischer’s sporadic simple group G1 = Fi23 described in [11]. He starts from an irreducible subgroup T1 of GL11(2) contained in T which is isomorphic to M23. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer’s original 3-transposition group Fi24 [6]. It is used here to show that G is isomorphic to the simple commutator subgroup Fi 24 of Fi24. We also determine a faithful permutation representation of G of degree 306936 with stabilizer G1 = 〈q, y, w〉 ∼= Fi23. It enabled MAGMA to calculate the character table of G automatically. Furthermore, we prove that G has two conjugacy classes of involutions z and u such that CG(u) = 〈q, y, t〉 ∼= 2Aut(Fi22). Moreover, we determine a presentation of H= CG(z) and a faithful permutation representation of degree 258048 for which we document a stabilizer.