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 centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting 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 group
Other 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 | ![]() |
1 | ![]() |
even;no | 1 |
2 + 1 + 1 + 1 + 1 + 1 + 1 | transposition, six fixed points | ![]() |
28 | ![]() ![]() |
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 |