# Symmetric group:S7

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## Contents

## Definition

This group is a finite group defined as the symmetric group on a set of size . The set is typically taken to be .

In particular, it is a symmetric group on finite set as well as a symmetric group of prime degree.

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 5040 | groups with same order | The order is |

exponent of a group | 420 | groups with same order and exponent of a group | groups with same exponent of a group | The exponent is the least common multiple of |

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

## Elements

`Further information: element structure of symmetric group:S7`

### Upto conjugacy

Partition | Verbal description of cycle type | Representative element | Size of conjugacy class | [[Formula for size | Even or odd? If even, splits? If splits, real in alternating group? | Element orders |
---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 + 1 + 1 | seven fixed points | -- the identity element | 1 | even; no | 1 | |

2 + 1 + 1 + 1 + 1 + 1 | transposition, five fixed points | 21 | , also in this case | odd | 2 | |

3 + 1 + 1 + 1 + 1 | one 3-cycle, four fixed points | 70 | even; no | 3 | ||

4 + 1 + 1 + 1 | one 4-cycle, three fixed points | 210 | odd | 4 | ||

2 + 2 + 1 + 1 + 1 | two 2-cycles, three fixed points | 105 | even;no | 2 | ||

5 + 1 + 1 | one 5-cycle, two fixed points | 504 | even; no | 5 | ||

3 + 2 + 1 + 1 | one 3-cycle, one 2-cycle, two fixed points | 420 | odd | 6 | ||

6 + 1 | one 6-cycle, one fixed point | 840 | odd | 6 | ||

4 + 2 + 1 | one 4-cycle, one 2-cycle, one fixed point | 630 | even;no | 4 | ||

2 + 2 + 2 + 1 | three 2-cycles, one fixed point | 105 | odd | 2 | ||

3 + 3 + 1 | two 3-cycles, one fixed point | 280 | even;no | 3 | ||

3 + 2 + 2 | one 3-cycle, two transpositions | 210 | even;no | 6 | ||

5 + 2 | one 5-cycle, one transposition | 504 | odd | 10 | ||

4 + 3 | one 4-cycle, one 3-cycle | 420 | odd | 12 | ||

7 | one 7-cycle | 720 | even;yes;no | 7 |

## GAP implementation

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

SymmetricGroup(7) |
SymmetricGroup |