Build a Sporadic Group in your Basement

Abstract

All simple finite groups are classified as members of specific families. With one exception, these families are infinite collections of groups sharing similar structures. The exceptional family of sporadic groups contains exactly twenty-six members. The five Mathieu groups are the most accessible of these sporadic cases. In this article, we explore connections between Mathieu groups and error-correcting communication codes. These connections permit simple, visual representations of the three largest Mathieu groups: M24, M23, and M22. Along the way, we provide a brief, nontechnical introduction to the field of coding theory.