# Mathieu group:M12

## Contents

## Definition

This group is defined as the Mathieu group of degree . It is one of the five simple Mathieu groups. It is given by the following permutation representation, as a subgroup of symmetric group:S12 as follows. Consider the following set of size 12: the projective line over field:F11. Explicitly, this set can be written as:

We have projective special linear group:PSL(2,11), acting naturally on this set as fractional linear transformations by:

This defines an embedding of inside .

The group is defined as the subgroup of generated by the image of and the permutation given by , which as a permutation is:

Note that this permutation fixes the points .

Note further that since all generating permutations are even permutations, this group is in fact a subgroup of alternating group:A12.

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 95040 | 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 |

## 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:M12`

## Linear representation theory

`Further information: linear representation theory of Mathieu group:M12`

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

degrees of irreducible representations over a splitting field (such as or ) | 1,11,11,16,16,45,54,55,55,55,66,99,120,144,176 grouped form (occurs once by default): 1, 11 (2 times), 16 (2 times), 45, 54, 55 (3 times), 66, 99, 120, 144, 176 maximum: 176, quasirandom degree: 11, number: 15, sum of squares: 95040 |

## GAP implementation

Unfortunately, GAP's SmallGroup library is not available for this order of group (95040) so the group cannot be constructed that way. It can be constructed in other ways:

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

MathieuGroup(12) |
MathieuGroup |

PerfectGroup(95040) or equivalently PerfectGroup(95040,1) |
PerfectGroup |

The group is somewhat cumbersome to manipulate directly because of its large size. Information about the group, including its character table, can be accessed using the symbol "M12" -- see linear representation theory of Mathieu group:M12#GAP implementation for more.