Element structure of symmetric group:S6
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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 is NOT isomorphic to projective general linear group:PGL(2,9). For proof of the non-isomorphism, see PGL(2,9) is not isomorphic to S6. 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 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 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 | ![]() |
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 rows = 11 conjugacy classes) | -- | -- | -- | 720 (equals order of the whole group) | -- | 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 |
Total (8 classes) | -- | -- | 11 | 720 | -- |