Element structure of symmetric group:S4

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This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S4.
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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 $4$.

Summary

Item	Value
order of the whole group (total number of elements)	24
conjugacy class sizes	1,3,6,6,8 maximum: 8, number of conjugacy classes: 5
order statistics	1 of order 1, 9 of order 2, 8 of order 3, 6 of order 4 maximum: 4, lcm (exponent of the whole group): 12

Family contexts

Note that if you go to the #Conjugacy class structure section of this article, you'll find a discussion of the conjugacy class structure with each of the below family interpretations.

Family name	Parameter values	General discussion of element structure of family
symmetric group	degree $n=4$	element structure of symmetric groups
projective general linear group of degree two over a finite field	field:F3, i.e., the group is $PGL(2,3)$	element structure of projective general linear group of degree two over a finite field
general affine group of degree two over a finite field	field:F2, i.e., the group is $GA(2,2)$	element structure of general affine group of degree two over a finite field

COMPARE AND CONTRAST: View element structure of groups of order 24 to compare and contrast the element structure with other groups of order 24.

Elements

Note that for elements whose order is not 2, the matrix for the right action is obtained by taking the transpose of the matrix for the left action.

Cycle decomposition notation	One-line notation, i.e., image of string $1,2,3,4$	Matrix (left action)
$()$	1234	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(3,4)$	1243	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ \end{array}\right)$
$(2,3)$	1324	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(2,3,4)$	1342	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \end{array}\right)$
$(2,4,3)$	1423	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ \end{array}\right)$
$(2,4)$	1432	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ \end{array}\right)$
$(1,2)$	2134	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(1,2)(3,4)$	2143	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ \end{array}\right)$
$(1,2,3)$	2314	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(1,2,3,4)$	2341	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ \end{array}\right)$
$(1,2,4,3)$	2413	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ \end{array}\right)$
$(1,2,4)$	2431	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ \end{array}\right)$
$(1,3,2)$	3124	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(1,3,4,2)$	3142	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ \end{array}\right)$
$(1,3)$	3214	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right)$
$(1,3,4)$	3241	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ \end{array}\right)$
$(1,3)(2,4)$	3412	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ \end{array}\right)$
$(1,3,2,4)$	3421	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ \end{array}\right)$
$(1,4,3,2)$	4123	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ \end{array}\right)$
$(1,4,2)$	4132	$\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ \end{array}\right)$
$(1,4,3)$	4213	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ \end{array}\right)$
$(1,4)$	4231	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 1& 0& 0& 0\\ \end{array}\right)$
$(1,4,2,3)$	4312	$\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ \end{array}\right)$
$(1,4)(2,3)$	4321	$\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ \end{array}\right)$

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

The conjugacy class sizes are $1,3,6,6,8$.

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 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	Partition in grouped form	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	Formula calculating element order
1 + 1 + 1 + 1	1 (4 times)	four cycles of size one each, i.e., four fixed points	$()$ -- the identity element	1	$\frac{4!}{(1{)}^4(4!)}$	even; no	1	$\mathrm{l}\mathrm{c}\mathrm{m}\{1,1,1,1\}$
2 + 1 + 1	2 (1 time), 1 (2 times)	one transposition (cycle of size two), two fixed points	$(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$, $(3,4)$	6	$\frac{4!}{[(2{)}^1(1!)][(1{)}^2(2!)]}$, also $(\genfrac{}{}{0ex}{}{4}{2})$	odd	2	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,1,1\}$
2 + 2	2 (2 times)	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	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,2\}$
3 + 1	3 (1 time), 1 (1 time)	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	$\frac{4!}{[(3{)}^1(1!)][(1{)}^1(1!)]}$ or $\frac{4!}{(3)(1)}$	even; yes; no	3	$\mathrm{l}\mathrm{c}\mathrm{m}\{3,1\}$
4	4 (1 time)	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	$\frac{4!}{(4{)}^1(1!)}$ or $\frac{4!}{4}$	odd	4	$\mathrm{l}\mathrm{c}\mathrm{m}\{4\}$
Total (5 rows)	--	--	--	24 (equals order of the whole group)	--	odd: 12 (2 classes) even; no: 4 (2 classes) even; yes; no: 8 (1 class)	order 1: 1 (1 class) order 2: 9 (2 classes) order 3: 8 (1 class) order 4: 6 (1 class)	--

Here is more information on the conjugacy classes:

FACTS TO CHECK AGAINST ON FIXED POINTS AND CYCLES
Fixed points: probability distribution of number of fixed points of permutations | expected number of fixed points of permutation equals one
Number of cycles: probability distribution of number of cycles of permutations | expected number of cycles of permutation equals harmonic number of degree

Partition	Number of elements in conjugacy class	Order of elements	Number of fixed points	Number of cycles (including fixed points)	Minimum number of transpositions that must be multiplied to obtain this cycle decomposition
1 + 1 + 1 + 1	1	1	4	4	0
2 + 1 + 1	6	2	2	3	1
2 + 2	3	2	0	2	2
3 + 1	8	3	1	2	2
4	6	4	0	1	3
Mean over conjugacy classes	24/5	8/5	7/5	6	8/5
Mean over elements	73/12	67/24	1	25/12	23/12

The mean over elements of the number of fixed points is $1$ for all symmetric groups on finite sets. The mean over elements of the number of cycles is $1+1/2+1/3+\dots +1/n$, which in this case is $1+1/2+1/3+1/4$.

For characters, see linear representation theory of symmetric group:S4.

Interpretation as projective general linear group of degree two

The symmetric group $S_4$ is isomorphic to $PGL(2,3)$, i.e., the projective general linear group of degree two over field:F3. Compare with element structure of projective general linear group of degree two over a finite field.

Nature of conjugacy class upstairs in $GL(2,q)$ (here $q=3$)	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class (generic odd $q$)	Size of conjugacy class ($q=3$)	Number of such conjugacy classes (generic odd $q$)	Number of such conjugacy classes ($q=3$)	Total number of elements (generic odd $q$)	Total number of elements ($q=3$)	Representatives of conjugacy classes as permutations
Diagonalizable over ${\mathbb{F}}_q$ with equal diagonal entries, hence a scalar	$\{a,a\}$ where $a\in {\mathbb{F}}_q^{\ast }$	$(x-a{)}^2$ where $a\in {\mathbb{F}}_q^{\ast }$	$x-a$ where $a\in {\mathbb{F}}_q^{\ast }$	1	1	1	1	1	1	$()$
Diagonalizable over ${\mathbb{F}}_{q^2}$, not over ${\mathbb{F}}_q$, eigenvalues are negatives of each other.	Pair of mutually negative conjugate elements of ${\mathbb{F}}_{q^2}$. All such pairs identified.	$x^2-\mu$, $\mu$ a nonzero non-square	Same as characteristic polynomial	$q(q-1)/2$	3	1	1	$q(q-1)/2$	3	$(1,2)(3,4)$
Diagonalizable over ${\mathbb{F}}_q$ with mutually negative diagonal entries.	$\{\lambda ,-\lambda \}$, all such pairs identified.	$x^2-{\lambda }^2$, all identified	Same as characteristic polynomial	$q(q+1)/2=(q^2+q)/2$	6	1	1	$q(q+1)/2=(q^2+q)/2$	6	$(1,2)$
Diagonalizable over ${\mathbb{F}}_{q^2}$, not over ${\mathbb{F}}_q$, eigenvalues are not negatives of each other.	Pair of conjugate elements of ${\mathbb{F}}_{q^2}$. Each pair identified with anything obtained by multiplying both elements of it by an element of ${\mathbb{F}}_q$.	$x^2-ax+b$, $a\ne 0$, irreducible; with identification.	Same as characteristic polynomial	$q(q-1)$	6	$(q-1)/2$	1	$q(q-1{)}^2/2=(q^3-2q^2+q)/2$	6	$(1,2,3,4)$
Not diagonal, has Jordan block of size two	$a\in {\mathbb{F}}_q^{\ast }$ (multiplicity 2). Each conjugacy class has one representative of each type.	$(x-a{)}^2$	Same as characteristic polynomial	$q^2-1$	8	1	1	$q^2-1$	8	$(1,2,3)$
Diagonalizable over ${\mathbb{F}}_q$ with distinct diagonal entries whose sum is not zero.	$\lambda ,\mu$ where $\lambda ,\mu \in {\mathbb{F}}_q^{\ast }$ and $\lambda +\mu \ne 0$. The pairs $\{\lambda ,\mu \}$ and $\{a\lambda ,a\mu \}$ are identified.	$x^2-(\lambda +\mu )x+\lambda \mu$, again with identification.	Same as characteristic polynomial.	$q(q+1)$	12	$(q-3)/2$	0	$q(q+1)(q-3)/2=(q^3-2q^2-3q)/2$	0	--
Total	NA	NA	NA	NA	NA	$q+2$	5	$q^3-q$	24	--

Convolution algebra on conjugacy classes

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Cayley graphs

With generating set all transpositions

Here, the generating set is the set of all transpositions. Since the generating set is a conjugacy class of involutions, the left and right Cayley graphs are identical. Further, we can unambiguously give a direction to each edge (away from the identity element) because there are no cycles of odd length, which follows from the fact that all elements of the generating set are odd permutations.

The following is some useful tabulated information about the Cayley graph. The edges to/from listed here are the edges for any representative element, not the total across the conjugacy class. Note that the sum of edges to/from in each row is $6$, which is the number of generators used in the generating set.

Conjugacy class	Distance from identity in Cayley graph	Number of elements	Edges to/from $()$	Edges to/from $(1,2)$-class	Edges to/from $(1,2,3)$-class	Edges to/from $(1,2)(3,4)$-class	Edges to/from $(1,2,3,4)$-class
$()$	0	1	0	6	0	0	0
$(1,2)$	1	6	1	0	4	1	0
$(1,2,3)$	2	8	0	3	0	0	3
$(1,2)(3,4)$	2	3	0	2	0	0	4
$(1,2,3,4)$	3	6	0	0	4	2	0

Bruhat ordering

The basic picture:

A fuller picture:

Basic tabulation by length

Length	Number of elements of that length	Elements of that length	Conjugacy class information for these elements
0	1	$()$ -- the identity element	forms a single conjugacy class
1	3	$(1,2)$, $(2,3)$, $(3,4)$	3 of 6 elements in conjugacy class of $(1,2)$
2	5	$(1,2)(3,4)$, $(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$	1 of 3 elements in conjugacy class of $(1,2)(3,4)$, 4 of 8 elements in conjugacy class of $(1,2,3)$
3	6	$(1,3)$, $(2,4)$, $(1,2,3,4)$, $(1,4,3,2)$, $(1,2,4,3)$, $(1,3,4,2)$	2 of 6 elements in conjugacy class of $(1,2)$, 4 of 6 elements in conjugacy class of $(1,2,3,4)$
4	5	$(1,3)(2,4)$, $(1,2,4)$, $(1,4,2)$, $(1,3,4)$, $(1,4,3)$	1 of 3 elements in conjugacy class of $(1,2)(3,4)$, 4 of 8 elements in conjugacy class of $(1,2,3)$
5	3	$(1,3,2,4)$, $(1,4,2,3)$, $(1,4)$	2 of 6 elements in conjugacy class of $(1,2,3,4)$, 1 of 6 elements in conjugacy class of $(1,4)$
6	1	$(1,4)(2,3)$	1 of 3 elements in conjugacy class of $(1,2)(3,4)$

Equivalence classes up to symmetries

These are equivalence classes up to the two symmetries: flipping the $s_i$s and a left-to-right order reversal symmetry. Elements in the same equivalence class are in the same conjugacy class in the group and the corresponding points in the Bruhat ordering are in the same orbit under automorphisms of the graph of the Bruhat ordering.

Elements in equivalence class	Number of elements	Length	Indegree	Outdegree	Image class under the anti-automorphism
$()$	1	0	0	1	$(1,4)(2,3)$
$(1,2)$, $(3,4)$	2	1	1	3	$(1,3,2,4)$, $(1,4,2,3)$
$(2,3)$	1	1	1	4	$(1,4)$
$(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$	4	2	2	3	$(1,2,4)$, $(1,3,4)$, $(1,4,2)$, $(1,4,3)$
$(1,2)(3,4)$	1	2	2	2	$(1,3)(2,4)$
$(1,3)$, $(2,4)$	2	3	2	2	$(1,3)$, $(2,4)$
$(1,2,3,4)$, $(1,4,3,2)$	2	3	2	2	$(1,2,3,4)$, $(1,4,3,2)$
$(1,2,4,3)$, $(1,3,4,2)$	2	3	3	3	$(1,2,4,3)$, $(1,3,4,2)$
$(1,2,4)$, $(1,3,4)$, $(1,4,2)$, $(1,4,3)$	4	4	3	2	$(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$
$(1,3)(2,4)$	1	4	2	2	$(1,2)(3,4)$
$(1,3,2,4)$, $(1,4,2,3)$	2	5	3	1	$(1,2)$, $(3,4)$
$(1,4)$	1	5	4	1	$(2,3)$
$(1,4)(2,3)$	1	6	1	0	$()$