# 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.

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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[{n_{1},n_{2},…}] | 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 |

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### 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 |

Intermediate sporadic simple groups.

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.

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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 |

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