# Named Groups

The Wolfram Language provides permutation representations for many important finite groups. Some of these groups are members of infinite families, parametrized by one or more integers; other groups are uniquely distinguished by their special properties and are frequently named after their discoverers.
This is the alternating group of degree 10:
Compute its order:
Give generators of an explicit permutation representation:
From these generators it is possible to reconstruct the group explicitly:
The Wolfram Language provides information on the following infinite families of groups, and on some groups not belonging to parametrized families.
 SymmetricGroup[n] symmetric group of degree n AlternatingGroup[n] alternating group of degree n CyclicGroup[n] cyclic group of order n DihedralGroup[n] dihedral group of the n-gon, of order 2n AbelianGroup[{n1,n2,…}] Abelian group isomorphic to a direct product of several cyclic groups
Named infinite families of groups.

### Mathieu Groups

The following five Mathieu groups were the first five sporadic simple groups to be discovered, in the second half of the nineteenth century, and are multiply transitive groups, all being subgroups of the largest one. The Wolfram Language provides default permutation representations for them.
 MathieuGroupM11 first Mathieu group, acting on 11 points MathieuGroupM12 second Mathieu group, acting on 12 points MathieuGroupM22 third Mathieu group, acting on 22 points MathieuGroupM23 fourth Mathieu group, acting on 23 points MathieuGroupM24 fifth Mathieu group, acting on 24 points
Mathieu groups.
These are the orders of the five sporadic simple Mathieu groups:
Explicit generators of a permutation representation of MathieuGroupM24 acting on 24 points:
To show that MathieuGroupM24 is 5-transitive, check transitivity of the group itself and of its first four stabilizers:
However, the stabilizer of five points is not transitive, because its action splits into two nontrivial orbits:
These are the orders of the stabilizers of the points of a base in the group. They correspond to the groups MathieuGroupM24, MathieuGroupM23, MathieuGroupM22, and then three more groups sometimes called Mathieu group , Mathieu group , and Mathieu group , which are not simple. Finally there is the cyclic group of order 3 and the trivial group:
A similar chain is obtained starting from MathieuGroupM12. It contains MathieuGroupM11 and non-simple groups that can be called Mathieu group , Mathieu group , and Mathieu group , with the trivial group at the end:
It is known that the largest permutation order in MathieuGroupM24 is 23. This is an example:

There are 26 sporadic simple groups (27 if the Tits group is included). Apart from the five Mathieu groups, the Wolfram Language provides permutation representations for those of intermediate support length. The largest ones are too big to be handled as permutation groups in practice, and it is more efficient to represent them as matrix groups. These are the 13 groups (including the Tits group) for which representations on domains of less than 50000 points are known.