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A classical group is a standard family of linear groups, principally the general/special linear groups and groups preserving nondegenerate alternating, symmetric, Hermitian or quadratic forms. Finite-field versions yield many finite simple groups after imposing the appropriate determinant or derived-subgroup condition and quotienting the scalar center of a group. Small dimensions and small fields have exceptional isomorphisms and nonsimple cases.

The classification of finite simple groups states that every finite simple group is isomorphic to a cyclic group of prime order, an alternating group $A_n$ with $n\ge 5$, a finite simple group of Lie type, or one of twenty-six sporadic simple groups. In the Lie-type families one takes the appropriate simple groups, respecting the exceptional small parameters and quotient constructions.^[ref]

For example, $C_7$ is a simple abelian group, while $A_5$ is a simple nonabelian group of order sixty. The symmetric group $S_3$ is not simple: its subgroup $A_3$ is a proper nontrivial normal subgroup. The theorem lists the simple building blocks that occur as composition factors. It does not classify all finite groups: different group extensions can assemble the same factors into nonisomorphic groups.

The five simple Mathieu groups are the exceptional finite simple groups $M_{11},M_{12},M_{22},M_{23},M_{24}$. Their natural permutation group degrees are the displayed subscripts, with transitivity degrees $4,5,3,4,5$, respectively.

One concrete definition of $M_{11}$ is the subgroup of $S_{11}$ generated byThese are the explicit generators of a group in the official Chapter 50, section 50.1-13, MathieuGroup example. Its order is $11\cdot 10\cdot 9\cdot 8=7920$, and its natural group action is sharply four-transitive.

The larger groups have useful design and coding realizations. $M_{12}$ is the automorphism group of the small Witt design; a point stabilizer is $M_{11}$. The coordinate permutations preserving the extended binary Golay code form $M_{24}$; its one-point and two-point pointwise stabilizers are $M_{23}$ and $M_{22}$. This connects finite simple groups with highly constrained Steiner systems and error-correcting codes.^[ref]