  In this article we give an self contained existence proof for J. Conway's
sporadic simple group Co_1 [4] using the second author's algorithm [14]
constructing finite simple groups from irreducible subgroups of GL_n(2). Here n
= 11 and the irreducible subgroup is the Mathieu group M_{24}. From the split
extension E of M_{24} by a uniquely determined 11-dimensional
GF(2)M_{24}-module V we construct the centralizer H = C_G(z) of a 2-central
involution z of E in an unknown target group G. Then we prove that all the
conditions of Algorithm 2.5 of [14] are satisfied. This allows us to construct
a simple subgroup G of GL_{276}(23) which we prove to be isomorphic with
Conway's original sporadic simple group Co_1 by means of a constructed faithful
permutation representation of G and Soicher's presentation [16] of the original
Conway group Co_1.
