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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 power degree.

## Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 40320 | groups with same order | The order is |

exponent of a group | 840 | groups with same order and exponent of a group "{{{" can not be assigned to a declared number type with value 3. | 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 "{{{" can not be assigned to a declared number type with value 3. | groups with same Frattini length |

## Elements

`Further information: element structure of symmetric group:S8`

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

2 + 1 + 1 + 1 + 1 + 1 + 1 | transposition, six fixed points | 28 | , also | odd | 2 | |

3 + 1 + 1 + 1 + 1 + 1 | one 3-cycle, five fixed points | 112 | even;no | 3 | ||

4 + 1 + 1 + 1 + 1 | one 4-cycle, four fixed points | 420 | odd | 4 | ||

2 + 2 + 1 + 1 + 1 + 1 | two transpositions, four fixed points | 210 | even;no | 2 | ||

5 + 1 + 1 + 1 | one 5-cycle, three fixed points | 1344 | even;no | 5 | ||

3 + 2 + 1 + 1 + 1 | one 3-cycle, one transposition, three fixed points | 1120 | odd | 6 | ||

6 + 1 + 1 | one 6-cycle, two fixed points | 3360 | odd | 6 | ||

4 + 2 + 1 + 1 | one 4-cycle, one 2-cycle, two fixed points | 2520 | even;no | 4 | ||

2 + 2 + 2 + 1 + 1 | three 2-cycles, two fixed points | 420 | odd | 2 | ||

3 + 3 + 1 + 1 | two 3-cycles, two fixed points | 1120 | even;no | 3 | ||

7 + 1 | one 7-cycle, one fixed point | 5760 | even;yes;no | 7 | ||

3 + 2 + 2 + 1 | one 3-cycle, two transpositions, one fixed point | 1680 | even;no | 6 | ||

4 + 3 + 1 | one 4-cycle, one 3-cycle, one fixed point | 3360 | odd | 12 | ||

5 + 2 + 1 | one 5-cycle, one 2-cycle, one fixed point | 4032 | odd | 10 | ||

2 + 2 + 2 + 2 | four 2-cycles | 105 | even;no | 2 | ||

4 + 2 + 2 | one 4-cycle, two 2-cycles | 1260 | odd | 4 | ||

3 + 3 + 2 | two 3-cycles, one 2-cycle | 1120 | odd | 6 | ||

6 + 2 | one 6-cycle, one 2-cycle | 3360 | even;no | 6 | ||

5 + 3 | one 5-cycle, one 3-cycle | 2688 | even;yes;no | 15 | ||

4 + 4 | two 4-cycles | 1260 | even;no | 4 | ||

8 | one 8-cycle | 5040 | odd | 8 |

## GAP implementation

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

SymmetricGroup(8) |
SymmetricGroup |