Element structure of symmetric group:S6

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S6.
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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 $\{1,2,3,4,5,6\}$.

Conjugacy class structure

As symmetric group

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	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	six fixed points	$()$ -- the identity element	1	$\frac{6!}{(1{)}^6(6!)}$	even; no	1
2 + 1 + 1 + 1 + 1	transposition, four fixed points	$(1,2)$	15	$\frac{6!}{(2)(1{)}^4(4!)}$	odd	2
3 + 1 + 1 + 1	one 3-cycle, three fixed points	$(1,2,3)$	40	$\frac{6!}{(3)(1{)}^3(3!)}$	even; no	3
4 + 1 + 1	one 4-cycle, two fixed points	$(1,2,3,4)$	90	$\frac{6!}{(4)(1{)}^2(2!)}$	odd	4
5 + 1	one 5-cycle, one fixed point	$(1,2,3,4,5)$	144	$\frac{6!}{(5)(1)}$	even; yes; yes	5
3 + 2 + 1	one 3-cycle, one 2-cycle, one fixed point	$(1,2,3)(4,5)$	120	$\frac{6!}{(3)(2)(1)}$	odd	6
2 + 2 + 1 + 1	double transposition: two 2-cycles, two fixed points	$(1,2)(3,4)$	45	$\frac{6!}{(2{)}^2(2!)(1{)}^2(2!)}$	even; no	2
2 + 2 + 2	triple transposition	$(1,2)(3,4)(5,6)$	15	$\frac{6!}{(2{)}^3(3!)}$	odd	2
4 + 2	one 4-cycle, one 2-cycle	$(1,2,3,4)(5,6)$	90	$\frac{6!}{(4)(2)}$	even; no	4
3 + 3	two 3-cycles	$(1,2,3)(4,5,6)$	40	$\frac{6!}{(3{)}^2(2!)}$	even; no	3
6	one 6-cycle	$(1,2,3,4,5,6)$	120	$\frac{6!}{6}$	odd	6

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	$(1,2)$, $(1,2)(3,4)(5,6)$	15	2	30	2
3 + 1 + 1 + 1, 3 + 3	$(1,2,3)$, $(1,2,3)(4,5,6)$	40	2	80	3
4 + 1 + 1, 4 + 2	$(1,2,3,4)$, $(1,2,3,4)(5,6)$	90	2	180	4
5 + 1	$(1,2,3,4,5)$	144	1	144	5
3 + 2 + 1, 6	$(1,2,3)(4,5)$, $(1,2,3,4,5,6)$	120	2	240	6
2 + 2 + 1 + 1	$(1,2)(3,4)$	45	1	45	2