# Mathieu group:M10

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## Contents

## Definition

This group, denoted , is the Mathieu group of degree . Explicitly, it can be defined as the isotropy subgroup (stabilizer) of two points under the natural action of Mathieu group:M12 on the projective line over field:F11. Since the action is a 2-transitive group action, it does not matter which two points we choose, but for concreteness, we can choose and .

Although it is one of the Mathieu groups, it is not among the five sporadic simple Mathieu groups. Rather, it is among the two non-simple Mathieu groups. The other is Mathieu group:M9. The Mathieu group is isomorphic to projective special linear group:PSL(3,4) and is hence simple but not a sporadic simple group.

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 720 | groups with same order | As Mathieu group : |

exponent of a group | 120 | 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 | derived subgroup is alternating group:A6. |

simple group | No |

## Subgroups

`Further information: subgroup structure of Mathieu group:M10`

### Quick summary

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

number of subgroups | 793 |

number of conjugacy classes of subgroups | 25 |

number of automorphism classes of subgroups | 24 |

isomorphism classes of Sylow subgroups and the corresponding Sylow numbers | 2-Sylow: semidihedral group:SD16, Sylow number is 45 3-Sylow: elementary abelian group:E9, Sylow number is 10 5-Sylow: cyclic group:Z5, Sylow number is 36 |

Hall subgroups | No Hall subgroups other than the trivial subgroup, whole group, and Sylow subgroups |

maximal subgroups | maximal subgroups have order 16, 20, 72, 360 |

normal subgroups | the only proper nontrivial normal subgroup is of index two and is isomorphic to alternating group:A6 (order 360) |

## Linear representation theory

`Further information: linear representation theory of Mathieu group:M10`

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

degrees of irreducible representations over a splitting field (such as or ) | 1, 1, 9, 9, 10, 10, 10, 16 grouped form: 1 (2 times), 9 (2 times), 10 (3 times), 16 (1 time) maximum: 16, number: 8, lcm: 720, sum of squares: 720 |

## GAP implementation

### Group ID

This finite group has order 720 and has ID 765 among the groups of order 720 in GAP's SmallGroup library. For context, there are 840 groups of order 720. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(720,765)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(720,765);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [720,765]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

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

MathieuGroup(10) |
MathieuGroup |