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Published online by Cambridge University Press: 31 October 2024
Properties of the Steiner system S(5, 8, 24) are given including the important Todd triangle that reveals the manner in which the 759 8-element subsets of the system (the octads) intersect one another. The Mathieu group M24 is the group of all permutations of the 24 letters Ω that preserve these 759 octads. We introduce the Miracle Octad Generator or MOG, a device in which the 759 octads are easily recognized, and explain where it comes from. A mnemonic for recovering the standard MOG labelling is given, along with examples of octad finding: that is to say, identifying the unique octad containing any given five points of the 24.
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