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2-graph 2-modular 2-transitive group 23 blocks 3-modular 50 points acting 2-transitively block design blocks of imprimitivity Brian Mortimer Carleton central involution characters of degree complete the proof composition factors composition series constructed Conway group CONWAYS THIRD GROUP denote design with parameters dimension 22 dimension 49 dimensional distinct blocks dual exactly factors of dimension failing to contain field of characteristic full automorphism group group C acting group C_ group H Higman-Sims group HS-module implies inner product intersect invariant irreducible characters irreducible representations J-module KQ spanned Lemma maximal subgroup MODULAR PERMUTATION REPRESENTATIONS Moreover orbits on Q order of G orthogonal pair of blocks permutation character permutation group permutation module KQ projection proof of Theorem rank REPRESENTATIONS OF CONWAYS Section self-dual set Q structure of KQ submodule of KTT subset subspace spanned sum of irreducible third possibility triples trivial module vectors 4e 4e Witt design