# Mathieu group:M11

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## Contents

## Definition

### In terms of

This group, termed the **Mathieu group of degree eleven** and denoted is the subgroup of the symmetric group of degree eleven defined as the isotropy subgroup of any point under the natural action of Mathieu group:M12 on the projective line over field:F11.

is in fact a subgroup of the alternating group of degree eleven.

### Relation with Mathieu groups

This is one of the five simple Mathieu groups, which form a subset of the sporadic simple groups. The parameters for the simple Mathieu groups are . There are also Mathieu groups for parameters , but these are not simple groups. The Mathieu group for parameter is a simple group that is not a sporadic simple group; it is isomorphic to the projective special linear group:PSL(3,4).

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 7920 | groups with same order | As Mathieu group : |

exponent of a group | 1320 | groups with same order and exponent of a group | groups with same exponent of a group | |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length |

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group | Yes | |

minimal simple group | No |

## Subgroups

`Further information: subgroup structure of Mathieu group:M11`

### Quick summary

Item | Value |
---|---|

number of subgroups | 8651 |

number of conjugacy classes of subgroups | 39 |

isomorphism classes of Sylow subgroups and corresponding Sylow numbers | 2-Sylow: semidihedral group:SD16, Sylow number is 495 3-Sylow: elementary abelian group:E9, Sylow number is 55 5-Sylow: cyclic group:Z5, Sylow number is 396 11-Sylow: cyclic group:Z11, Sylow number is 144 |

Hall subgroups | apart from the whole group, trivial subgroup, and Sylow subgroups, there exist the following Hall subgroups: -Hall (order 144), -Hall (order 55), -Hall (order 720) |

maximal subgroups | maximal subgroups of order 48, 120, 144, 660, 720 |

normal subgroups | the group is simple non-abelian, so the only normal subgroups are the whole group and the trivial subgroup |

subgroups that are simple non-abelian groups (apart from the whole group) | alternating group:A5 (order 60), alternating group:A6 (order 360), projective special linear group:PSL(2,11) (order 660) |

## Linear representation theory

`Further information: linear representation theory of Mathieu group:M11`

### Summary

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (such as or ) | 1,10,10,10,11,16,16,44,45,55 grouped form: 1 (1 time), 10 (3 times), 11 (1 time), 16 (2 times), 44 (1 time), 45 (1 time), 55 (1 time) maximum: 55, quasirandom degree: 10, number: 10, lcm: 7920, sum of squares: 7920 |

## GAP implementation

The Mathieu group has order 7920. Unfortunately, GAP's SmallGroup library is not available for this order. The group can be constructed in either of these ways:

Description | Functions used |
---|---|

MathieuGroup(11) |
MathieuGroup |

PerfectGroup(7920) or equivalently PerfectGroup(7920,1) |
PerfectGroup |