# Element structure of symmetric group:S8

From Groupprops

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S8.

View element structure of particular groups | View other specific information about symmetric group:S8

This article describes the element structure of symmetric group:S8.

See also element structure of symmetric groups.

For convenience, we take the underlying set to be .

## 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 8.

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, two 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 |