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3 - The Miracle Octad Generator

Published online by Cambridge University Press:  31 October 2024

Robert T. Curtis
Robert T. Curtis
Affiliation:
University of Birmingham
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Summary

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.

Keywords

S(5, 8, 24)Todd’s triangleMiracle Octad Generatorstandard labelling
Type
Chapter
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The Art of Working with the Mathieu Group M24 , pp. 17 - 27
Publisher: Cambridge University Press
Print publication year: 2024

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  • The Miracle Octad Generator
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
  • Chapter DOI: https://doi.org/10.1017/9781009405683.005
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  • The Miracle Octad Generator
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
  • Chapter DOI: https://doi.org/10.1017/9781009405683.005
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • The Miracle Octad Generator
  • Robert T. Curtis, University of Birmingham
  • Book: The Art of Working with the Mathieu Group M24
  • Online publication: 31 October 2024
  • Chapter DOI: https://doi.org/10.1017/9781009405683.005
Available formats
×