Element structure of symmetric group:S4

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S3.
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This article discusses symmetric group:S4, the symmetric group of degree four. We denote its elements as acting on the set $\{1,2,3.4\}$, written using cycle decompositions, with composition by function composition where functions act on the left.

Since this group is a complete group (i.e., every automorphism is inner and the center is trivial), the classification of elements up to conjugacy is the same as the classification up to automorphisms. Further, since cycle type determines conjugacy class for symmetric groups, the conjugacy classes are parametrized by cycle types, which in turn are parametrized by unordered integer partitions of $3$.

To put this in context, refer element structure of symmetric group:S4.

Conjugacy class structure

Summary

For any symmetric group, cycle type determines conjugacy class, i.e., the cycle type of a permutation (which describes the sizes of the cycles in a cycle decomposition of that permutation), determines its conjugacy class. In other words, two permutations are conjugate if and only if they have the same number of cycles of each size.

The cycle types (and hence the conjugacy classes) are parametrized by partitions of the size of the set. We describe the situation for this group:

Partition	Verbal description of cycle type	Elements with the cycle type	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element order
1 + 1 + 1 + 1	four cycles of size one each, i.e., four fixed points	$()$ -- the identity element	1	$\!{\frac {4!}{(1)^{4}(4!)}}$	even; no	1
2 + 1 + 1	one transposition (cycle of size two), two fixed points	$(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$, $(3,4)$	6	$\!{\frac {4!}{(2)(1)^{2}(2!)}}$	odd	2
2 + 2	double transposition: two cycles of size two	$(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$	3	$\!{\frac {4!}{(2)^{2}(2!)}}$	even; no	2
3 + 1	one 3-cycle, one fixed point	$(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$, $(3,4,1)$, $(3,1,4)$, $(4,1,2)$, $(4,2,1)$	8	$\!{\frac {4!}{(3)(1)}}$	even; yes; no	3
4	one 4-cycle, no fixed points	$(1,2,3,4)$, $(1,2,4,3)$, $(1,3,2,4)$, $(1,3,4,2)$, $(1,4,2,3)$, $(1,4,3,2)$	6	$\!{\frac {4!}{4}}$	odd	4