# Alternating group:A6

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## Contents

## Definition

The alternating group is defined in the following equivalent ways:

- It is the group of even permutations (viz., the alternating group) on six elements.
- It is the projective special linear group .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 360#Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 360 | groups with same order | As : As , |

exponent of a group | 60 | groups with same order and exponent of a group | groups with same exponent of a group | Elements of order . |

derived length | -- | -- | not a solvable group. |

nilpotency class | -- | -- | not a nilpotent group. |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | Frattini-free group: intersection of maximal subgroups is trivial. |

minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | |

subgroup rank of a group | 2 | groups with same order and subgroup rank of a group | groups with same subgroup rank of a group | -- |

max-length of a group | 5 | groups with same order and max-length of a group | groups with same max-length of a group | -- |

## Group properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

Abelian group | No | , don't commute | is non-abelian, . |

Nilpotent group | No | Centerless: The center is trivial | is non-nilpotent, . |

Metacyclic group | No | Simple and non-abelian | is not metacyclic, . |

Supersolvable group | No | Simple and non-abelian | is not supersolvable, . |

Solvable group | No | is not solvable, . | |

Simple non-abelian group | Yes | alternating groups are simple, projective special linear group is simple | |

T-group | Yes | Simple and non-abelian | |

Ambivalent group | Yes | Classification of ambivalent alternating groups | |

Rational-representation group | No | ||

Rational group | No | ||

Complete group | No | Conjugation by odd permutations in gives outer automorphisms. |

## Elements

`Further information: element structure of alternating group:A6`

## Subgroups

`Further information: subgroup structure of alternating group:A6`

## Supergroups

`Further information: supergroups of alternating group:A6`

### Subgroups: making some or all the outer automorphisms inner

All the groups listed here are almost simple groups, because alternating group:A6 is a simple non-abelian group.

The outer automorphism group of alternating group:A6 is a Klein four-group. In particular, it has order 4. By the fourth isomorphism theorem, subgroups of the automorphism group containing the inner automorphism group correspond to subgroups of the outer automorphism group, which is the quotient group of the automorphism group by the inner automorphism group.

Since alternating group:A6 is a centerless group, it is identified naturally with its inner automorphism group, so each of the subgroups of the automorphism group containing the inner automorphism group is *also* a group containing as a self-centralizing normal subgroup. The whole automorphism group contains as a NSCFN-subgroup.

Below is the complete list of these groups.:

Group containing and contained in (this is an almost simple group) | Corresponding subgroup of viewed as Klein four-group | Order of group | Order of corresponding subgroup of = index of in group = order of group/360 | Second part of GAP ID of big group (GAP ID is (order,2nd part)) |
---|---|---|---|---|

alternating group:A6 | trivial subgroup | 360 | 1 | 118 |

symmetric group:S6 | one of the three copies of Z2 in V4 | 720 | 2 | 763 |

projective general linear group:PGL(2,9) | one of the three copies of Z2 in V4 | 720 | 2 | 764 |

Mathieu group:M10 | one of the three copies of Z2 in V4 | 720 | 2 | 765 |

automorphism group of alternating group:A6 | whole group | 1440 | 4 | 5841 |

### Quotients: Stem extensions and Schur covering groups

All the groups listed here are quasisimple groups, because alternating group:A6 is a simple non-abelian group.

`Further information: Group cohomology of alternating group:A6#Schur multiplier, group cohomology of alternating groups`

The Schur multiplier of alternating group:A6 is cyclic group:Z6, i.e., the group . For each of the possible quotient groups of , there is a *unique* stem extension with that as base normal subgroup and alternating group:A6 as quotient. The stem extension for the whole Schur multiplier is the unique Schur covering group, also called the universal central extension.

Note that *uniqueness* follows from being a perfect group.

The list is below:

Group at base of stem extension | Order | Corresponding stem extension group (this is a quasisimple group) | Order | Second part of GAP ID (GAP ID is (order,second part)) |
---|---|---|---|---|

trivial group | 1 | alternating group:A6 | 360 | 118 |

cyclic group:Z2 | 2 | special linear group:SL(2,9), also denoted to indicate that it is a double cover of alternating group | 720 | 409 |

cyclic group:Z3 | 3 | triple cover of alternating group:A6 | 1080 | 260 |

cyclic group:Z6 | 6 | Schur cover of alternating group:A6 | 2160 | (ID not available for this order) |

## Linear representation theory

`Further information: linear representation theory of alternating group:A6`

### Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,5,5,8,8,9,10 grouped form: 1 (1 time), 5 (2 times), 8 (2 times), 9 (1 time), 10 (1 time) maximum: 10, lcm: 360, number: 7, sum of squares: 360 |

Ring generated by character values | |

Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) | Quadratic extension of Same as field generated by character values |

Orbits of irreducible representations under action of automorphism group | orbits of size 1 for representations of degree 1,9,10; orbits of size two for degree 5 and degree 8 representations (the degree 8 representations are interchanged under conjugation by an odd permutation; the degree 5 representations are interchaged by an automorphism that is outer for as well) |

Orbits of irreducible representations under action of Galois group | orbits of size 1 for representations of degree 1,5,5,9,10; orbit of size two for degree 8 representations (automorphism ) |

Minimal splitting field in prime characteristic | Case : prime field Case : quadratic extension of |

Smallest size splitting field | field:F11 |

Degrees of irreducible representations over the rational numbers | 1,5,5,9,10,16 |

### Character table

## GAP implementation

### Group ID

This finite group has order 360 and has ID 118 among the groups of order 360 in GAP's SmallGroup library. For context, there are 162 groups of order 360. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(360,118)`

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

`gap> G := SmallGroup(360,118);`

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

`IdGroup(G) = [360,118]`

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 |
---|---|

AlternatingGroup(6) |
AlternatingGroup |

PSL(2,9) |
PSL |