# Alternating group:A9

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## Contents

## Definition

This group is defined as the alternating group of degree 9, i.e., the alternating group on a set of size 9. This can be taken as the group on .

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 181440 | groups with same order | As : |

exponent of a group | 1260 | groups with same order and exponent of a group | groups with same exponent of a group |

## Linear representation theory

`Further information: linear representation theory of alternating group:A9`

### Summary

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

degrees of irreducible representations over a splitting field (such as or ) | 1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 maximum: 216, number: 18, sum of squares: 181440 |

minimal splitting field, i.e., smallest field of realization of all irreducible representations (characteristic zero) | where is a primitive fifteenth root of unity Same as Same as field generated by character values |

condition for a field of characteristic not 2,3,5,7 to be a splitting field | -15 should be a square in that field |

## GAP implementation

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

AlternatingGroup(9) |
AlternatingGroup |