View source for Element structure of symmetric group:S8

{{group-specific information|
group = symmetric group:S8|
information type = element structure|
connective = of}}

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 <math>\{ 1,2,3,4,5,6,7,8\}</math>.

==Conjugacy class structure==

{{conjugacy class structure facts to check against}}

===Interpretation as symmetric group===

{{symmetric group conjugacy class structure facts to check against}}

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.

<section begin="conjugacy class structure"/>

{| class="sortable" border="1"
! 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 || <math>()</math> -- the identity element || 1 || <math>\frac{8!}{(1)^8(8!)}</math> || even;no || 1
|-
| 2 + 1 + 1 + 1 + 1 + 1 + 1 || transposition, six fixed points || <math>(1,2)</math> || 28 || <math>\frac{8!}{(2)(1)^6(6!)}</math>, also <math>\binom{8}{2}</math> || odd || 2
|-
| 3 + 1 + 1 + 1 + 1 + 1 || one 3-cycle, five fixed points || <math>(1,2,3)</math> || 112 || <math>\frac{8!}{(3)(1)^5(5!)}</math> || even;no || 3
|-
| 4 + 1 + 1 + 1 + 1 || one 4-cycle, four fixed points || <math>(1,2,3,4)</math> || 420 || <math>\frac{8!}{(4)(1)^4(4!)}</math> || odd || 4
|-
| 2 + 2 + 1 + 1 + 1 + 1 || two transpositions, four fixed points || <math>(1,2)(3,4)</math> || 210 || <math>\frac{8!}{(2)^2(2!)(1)^4(4!)}</math> || even;no || 2
|-
| 5 + 1 + 1 + 1 || one 5-cycle, three fixed points || <math>(1,2,3,4,5)</math> || 1344  || <math>\frac{8!}{(5)(1)^3(3!)}</math> || even;no || 5
|-
| 3 + 2 + 1 + 1 + 1 || one 3-cycle, one transposition, three fixed points || <math>(1,2,3)(4,5)</math> || 1120 || <math>\frac{8!}{(3)(2)(1)^3(3!)}</math> || odd || 6
|-
| 6 + 1 + 1 || one 6-cycle, two fixed points || <math>(1,2,3,4,5,6)</math> || 3360 || <math>\frac{8!}{(6)(1)^2(2!}</math> || odd || 6
|-
| 4 + 2 + 1 + 1 || one 4-cycle, one 2-cycle, two fixed points || <math>(1,2,3,4)(5,6)</math> || 2520 || <math>\frac{8!}{(4)(2)(1)^2(2!)}</math> || even;no || 4
|-
| 2 + 2 + 2 + 1 + 1 || three 2-cycles, two fixed points || <math>(1,2)(3,4)(5,6)</math> || 420 || <math>\frac{8!}{(2)^3(3!)(1)^2(2!)}</math> || odd || 2
|-
| 3 + 3 + 1 + 1 || two 3-cycles, two fixed points || <math>(1,2,3)(4,5,6)</math> || 1120 || <math>\frac{8!}{(3)^2(2!)(1)^2(2!)}</math> || even;no || 3
|-
| 7 + 1 || one 7-cycle, one fixed point || <math>(1,2,3,4,5,6,7)</math> || 5760 || <math>\frac{8!}{(7)(1)}</math> || even;yes;no || 7
|-
| 3 + 2 + 2 + 1 || one 3-cycle, two transpositions, one fixed point || <math>(1,2,3)(4,5)(6,7)</math> || 1680 || <math>\frac{8!}{(3)(2)^2(2!)(1)}</math> || even;no || 6
|-
| 4 + 3 + 1 || one 4-cycle, one 3-cycle, one fixed point || <math>(1,2,3,4)(5,6,7)</math> || 3360 || <math>\frac{8!}{(4)(3)(1)}</math> || odd || 12
|-
| 5 + 2 + 1 || one 5-cycle, one 2-cycle, one fixed point || <math>(1,2,3,4,5)(6,7)</math> || 4032 || <math>\frac{8!}{(5)(2)(1)}</math> || odd || 10
|-
| 2 + 2 + 2 + 2 || four 2-cycles || <math>(1,2)(3,4)(5,6)(7,8)</math> || 105 || <math>\frac{8!}{(2)^4(4!)}</math> || even;no || 2
|-
| 4 + 2 + 2 || one 4-cycle, two 2-cycles || <math>(1,2,3,4)(5,6)(7,8)</math> || 1260 || <math>\frac{8!}{(4)(2)^2(2!)}</math> || odd || 4
|-
| 3 + 3 + 2 || two 3-cycles, one 2-cycle || <math>(1,2,3)(4,5,6)(7,8)</math> || 1120 || <math>\frac{8!}{(3)^2(2!)(2)}</math> || odd || 6
|-
| 6 + 2 || one 6-cycle, one 2-cycle || <math>(1,2,3,4,5,6)(7,8)</math> || 3360 || <math>\frac{8!}{(6)(2)}</math> || even;no || 6
|-
| 5 + 3 || one 5-cycle, one 3-cycle || <math>(1,2,3,4,5)(6,7,8)</math> || 2688 || <math>\frac{8!}{(5)(3)}</math> || even;yes;no || 15
|-
| 4 + 4 || two 4-cycles || <math>(1,2,3,4)(5,6,7,8)</math> || 1260 || <math>\frac{8!}{(4)^2(2!)}</math> || even;no || 4
|-
| 8 || one 8-cycle || <math>(1,2,3,4,5,6,7,8)</math> || 5040 || <math>\frac{8!}{8}</math> || odd || 8
|}

<section end="conjugacy class structure"/>