Book contents
- Frontmatter
- Dedication
- Contents
- Figures
- Tables
- Preface
- Acknowledgements
- 1 Introduction
- 2 Steiner Systems
- 3 The Miracle Octad Generator
- 4 The Binary Golay Code
- 5 Uniqueness of the Steiner System S(5, 8, 24) and the Group M24
- 6 The Hexacode
- 7 Elements of the Mathieu Group M24
- 8 The Maximal Subgroups of M24
- 9 The Mathieu Group M24
- 10 The Leech Lattice M24
- 11 The Conway Group ·O
- 12 Permutation Actions of M24
- 13 Natural Generators of the Mathieu Groups
- 14 Symmetric Generation Using M24
- 15 The Thompson Chain of Subgroups of Co1
- Appendix MAGMA Code for 7★36 : A9 ↦ Co1
- References
- Index
3 - The Miracle Octad Generator
Published online by Cambridge University Press: 31 October 2024
Book contents
- Frontmatter
- Dedication
- Contents
- Figures
- Tables
- Preface
- Acknowledgements
- 1 Introduction
- 2 Steiner Systems
- 3 The Miracle Octad Generator
- 4 The Binary Golay Code
- 5 Uniqueness of the Steiner System S(5, 8, 24) and the Group M24
- 6 The Hexacode
- 7 Elements of the Mathieu Group M24
- 8 The Maximal Subgroups of M24
- 9 The Mathieu Group M24
- 10 The Leech Lattice M24
- 11 The Conway Group ·O
- 12 Permutation Actions of M24
- 13 Natural Generators of the Mathieu Groups
- 14 Symmetric Generation Using M24
- 15 The Thompson Chain of Subgroups of Co1
- Appendix MAGMA Code for 7★36 : A9 ↦ Co1
- References
- Index
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.
- Type
- Chapter
- Information
- The Art of Working with the Mathieu Group M24 , pp. 17 - 27Publisher: Cambridge University PressPrint publication year: 2024