On 20 January 2015, Paul Hjelmstad posted the following question on the GAP forum:

Is there an easy way to generate a Steiner system S(5,8,24) for the Mathieu Group M_{24}, if a Steiner system S(5,6,12) for the Mathieu Group M_{12} is known?

I learned about the Mathieu groups when I was a student. I read Heinz Lüneburg’s book *Transitive Erweiterungen endlicher Permutationsgruppen* in the Springer Lecture Notes series, in which he constructed the Steiner system (and group) by extending three times the projective plane of order 4.

But, at about the same time, I attended a remarkable series of lectures by Graham Higman. In these lectures, he constructed the outer automorphism of the symmetric group S_{6}, and used it to construct and prove the uniqueness of the projective plane of order 4, the Moore graph of valency 7, and the Steiner system S(5,6,12), and hence deduce properties of their automorphism groups. He then gave an analogue of the “doubling” construction from S_{6} to M_{12}, starting with M_{12}, constructing its outer automorphism, and using this to build M_{24} and its Steiner system.

I don’t think he published any of this, and I doubt whether any notes are now available. I used the first part of the material in chapter 6 of my book *Designs, Graphs, Codes, and their Links* with Jack van Lint. The outer automorphism of S_{6} is described here.

So I have written out a description of the path from the small to the large Mathieu group, and put a copy here with my lecture notes. This document could be regarded as a supplement to Chapter 6 of my book with Jack. It may be of interest to someone else; let me know if you find it so.

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About Peter Cameron

I count all the things that need to be counted.