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.
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 |
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
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| MathieuGroupM12 |
second Mathieu group, acting on 12 points
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| MathieuGroupM22 |
third Mathieu group, acting on 22 points
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| MathieuGroupM23 |
fourth Mathieu group, acting on 23 points
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| MathieuGroupM24 |
fifth Mathieu group, acting on 24 points
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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:
, 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:
, 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:
Other Sporadic Simple Groups
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.
| HigmanSimsGroupHS | Higman–Sims sporadic simple group  |
| McLaughlinGroupMcL | McLaughlin sporadic simple group  |
| JankoGroupJ1 | Janko sporadic simple group  |
| JankoGroupJ2 | Janko sporadic simple group  |
| JankoGroupJ3 | Janko sporadic simple group  |
| ConwayGroupCo2 | Conway sporadic simple group  |
| ConwayGroupCo3 | Conway sporadic simple group  |
| SuzukiGroupSuz | Suzuki sporadic simple group  |
| HeldGroupHe | Held sporadic simple group  |
| RudvalisGroupRu | Rudvalis sporadic simple group  |
| FischerGroupFi22 | Fischer sporadic simple group  |
| FischerGroupFi23 | Fischer sporadic simple group  |
| TitsGroupT | Tits simple group  |
Some sporadic groups are related to symmetries of the Leech lattice, a particular lattice in a Euclidean 24-dimensional space. These are sometimes known as the "second generation" of the sporadic simple groups.
For example, these are generators for JankoGroupJ2 acting on 100 points:
This is a chain of stabilizers of ConwayGroupCo2, acting on 2300 points. The base has six points only, and hence knowing the images of these six points suffices to uniquely identify each permutation in the group:
These are other sporadic simple groups representable as permutation groups on less than 50000 points:
| ONanGroupON | O'Nan sporadic simple group  |
| HaradaNortonGroupHN | Harada–Norton sporadic simple group  |
| ConwayGroupCo1 | Conway sporadic simple group  |
| FischerGroupFi24Prime | Fischer sporadic simple group  |
| ThompsonGroupTh | Thompson sporadic simple group  |
| JankoGroupJ4 | Janko sporadic simple group  |
| LyonsGroupLy | Lyons sporadic simple group  |
| BabyMonsterGroupB | Baby monster sporadic simple group  |
| MonsterGroupM | Monster group  |