# Element structure of symmetric group:S6

From Groupprops

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S6.

View element structure of particular groups | View other specific information about symmetric group:S6

This article describes the element structure of symmetric group:S6.

See also element structure of symmetric groups.

For convenience, we take the underlying set to be .

This group isNOTisomorphic to projective general linear group:PGL(2,9). For the element structure of that group, see element structure of projective general linear group:PGL(2,9).

## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Interpretation as symmetric group

FACTS TO CHECK AGAINST SPECIFICALLY FOR SYMMETRIC GROUPS AND ALTERNATING GROUPS:

Please read element structure of symmetric groups for a summary description.Conjugacy class parametrization: cycle type determines conjugacy class (in symmetric group)Conjugacy class sizes: conjugacy class size formula in symmetric groupOther facts: even permutation (definition) -- the alternating group is the set of even permutations | splitting criterion for conjugacy classes in the alternating group (from symmetric group)| criterion for element of alternating group to be real

For a symmetric group, cycle type determines conjugacy class, so the conjugacy classes are parametrized by the set of unordered integer partitions of the number 6.

Partition | Partition in grouped form | 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 | Formula calculating element order |
---|---|---|---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 + 1 | 1 (6 times) | six fixed points | -- the identity element | 1 | even; no | 1 | ||

2 + 1 + 1 + 1 + 1 | 2 (1 time), 1 (4 times) | transposition, four fixed points | 15 | odd | 2 | |||

3 + 1 + 1 + 1 | 3 (1 time), 1 (3 times) | one 3-cycle, three fixed points | 40 | even; no | 3 | |||

4 + 1 + 1 | 4 (1 time), 1 (2 times) | one 4-cycle, two fixed points | 90 | odd | 4 | |||

2 + 2 + 1 + 1 | 2 (2 times), 1 (2 times) | double transposition: two 2-cycles, two fixed points | 45 | even; no | 2 | |||

5 + 1 | 5 (1 time), 1 (1 time) | one 5-cycle, one fixed point | 144 | even; yes; yes | 5 | |||

3 + 2 + 1 | 3 (1 time), 2 (1 time), 1 (1 time) | one 3-cycle, one 2-cycle, one fixed point | 120 | odd | 6 | |||

2 + 2 + 2 | 2 (3 times) | triple transposition | 15 | odd | 2 | |||

4 + 2 | 4 (1 time), 2 (1 time) | one 4-cycle, one 2-cycle | 90 | even; no | 4 | |||

3 + 3 | 3 (2 times) | two 3-cycles | 40 | even; no | 3 | |||

6 | 6 (1 time) | one 6-cycle | 120 | odd | 6 | |||

Total (11) | -- | -- | -- | 720 | -- | odd: 360, 5 classes even;no: 216, 5 classes even;yes;yes: 144, 1 class |
order 1: 1, order 2: 75, order 3: 80, order 4: 180, order 5: 144, order 6: 240 | -- |

## Automorphism class structure

Partitions for cycle types in one automorphism class | Representative elements for each | Size of each conjugacy class | Number of classes | Total size | Element orders |
---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 + 1 | 1 | 1 | 1 | 1 | |

2 + 1 + 1 + 1 + 1, 2 + 2 + 2 | , | 15 | 2 | 30 | 2 |

3 + 1 + 1 + 1, 3 + 3 | , | 40 | 2 | 80 | 3 |

4 + 1 + 1 | 90 | 1 | 90 | 4 | |

4 + 2 | 90 | 1 | 90 | 4 | |

5 + 1 | 144 | 1 | 144 | 5 | |

3 + 2 + 1, 6 | , | 120 | 2 | 240 | 6 |

2 + 2 + 1 + 1 | 45 | 1 | 45 | 2 |