Sporadic simple group

A sporadic simple group is one of the twenty-six exceptional finite nonabelian simple groups that lie outside the alternating and Lie-type families in the classification of finite simple groups. “Sporadic” describes their place in that classification; it does not mean randomly chosen or lacking internal structure.^[ref]

For example, the Mathieu group $M_{24}$ acts by coordinate permutations preserving the extended binary Golay code on twenty-four positions. Thus an exceptional group can emerge as the exact symmetry of a highly organized error-correcting code. The Monster group is another sporadic example, vastly larger. In contrast, the alternating group $A_5$ is simple but belongs to an infinite family, so is not sporadic.