“There are almost as many different constructions of M24 as there have been mathematicians interested in that most remarkable of all finite groups”.

In this book the study of the Mathieu group M24 (and other Mathieu groups it contains) falls within the scope of what E. E. Shult called the Ivanov– Shpectorov theory of geometries. This theory has been developed to construct and identify large sporadic simple groups including the Baby Monster, the Fourth Janko Group J4 and the Monster. The most dramatic outcome of the theory was the proof of the famous Y -presentation conjecture for the Monster, which for a long time remained unobtainable by use of the other techniques. In the case of M24 the way in which the theory develops can be projected onto the familiar structures of the Steiner system on 24 points and the Golay code, thus presenting a bold illustration of the theory as well as providing a fresh look at familiar, nearly classical structures. I am extremely grateful to Madeleine Whybrow, William Giuliano and the anonymous referees for suggesting thoughtful corrections, clarifications and modifications after reading earlier versions of the book.