Quiz:Symmetric group:S4

See symmetric group:S4. We take the symmetric group on the set $\! \{1,2,3,4\}$ of size four.

Elements

See element structure of symmetric group:S4 for full details.

Element orders and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

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	$\operatorname{lcm}\{ 1,1,1,1 \}$
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 $\binom{4}{2}$	odd	2	$\operatorname{lcm}\{2,1,1 \}$
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	$\operatorname{lcm}\{2,2 \}$
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	$\operatorname{lcm}\{3,1 \}$
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	$\operatorname{lcm} \{ 4 \}$
Total (5 rows, 5 being the number of unordered integer partitions of 4)	--	--	--	24 (equals 4!, the 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)	--

1 What are the possible orders of elements of the symmetric group of degree four?

	1,2,3,4 only
	1,2,3,4,6 only
	1,2,3,4,8 only
	1,2,3,4,6,8 only
	1,2,3,4,6,8,12 only

2 For which of the following orders of elements is there more than one conjugacy class of elements with that order in the symmetric group of degree four?

	1
	2
	3
	4
	6
	8
	12

3 Which of the following is a representative of the conjugacy class in the symmetric group of degree four that has the largest size?

	$(1,2)$
	$(1,2,3)$
	$(1,2,3,4)$
	$(1,2)(3,4)$

Subgroups

See subgroup structure of symmetric group:S4 for background information.

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]

Quick summary

Item	Value
Number of subgroups	30 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups	11 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups	11 Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: dihedral group:D8 (order 8), Sylow number is 3, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 4, fusion system is non-inner fusion system for cyclic group:Z3
Hall subgroups	Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups
maximal subgroups	maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4).
normal subgroups	There are four normal subgroups: the whole group, the trivial subgroup, A4 in S4, and normal V4 in S4.

Table classifying subgroups up to automorphisms

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Automorphism class of subgroups	Representative	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
trivial subgroup	$\{ () \}$	trivial group	1	24	1	1	1	symmetric group:S4	1
S2 in S4	$\{ (), (1,2) \}$	cyclic group:Z2	2	12	1	6	6	--	--
subgroup generated by double transposition in S4	$\{ (), (1,2)(3,4) \}$	cyclic group:Z2	2	12	1	3	3	--	2
Z4 in S4	$\langle (1,2,3,4) \rangle$	cyclic group:Z4	4	6	1	3	3	--	--
normal Klein four-subgroup of S4	$\{ (), (1,2)(3,4),$ $(1,3)(2,4), (1,4)(2,3) \}$	Klein four-group	4	6	1	1	1	symmetric group:S3	1	2-core
non-normal Klein four-subgroups of S4	$\langle (1,2), (3,4) \rangle$	Klein four-group	4	6	1	3	3	--	--
D8 in S4	$\langle (1,2,3,4), (1,3) \rangle$	dihedral group:D8	8	3	1	3	3	--	--	2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8
A3 in S4	$\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	8	1	4	4	--	--	3-Sylow, fusion system is non-inner fusion system for cyclic group:Z3
S3 in S4	$\langle (1,2,3), (1,2) \rangle$	symmetric group:S3	6	4	1	4	4	--	--
A4 in S4	$\langle (1,2,3), (1,2)(3,4) \rangle$	alternating group:A4	12	2	1	1	1	cyclic group:Z2	1
whole group	$\langle (1,2,3,4), (1,2) \rangle$	symmetric group:S4	24	1	1	1	1	trivial group	0
Total (11 rows)	--	--	--	--	11	--	30	--	--	--

1 What are the possible orders of subgroups of the symmetric group of degree four?

	1,2,6,24 only
	1,3,8,24 only
	1,4,12,24 only
	1,2,3,4,12,24 only
	1,2,3,4,6,12,24 only
	1,2,3,4,8,12,24 only
	1,2,3,4,6,8,12,24 only

2 What are the possible orders of normal subgroups of the symmetric group of degree four?

	1,2,6,24 only
	1,3,8,24 only
	1,4,12,24 only
	1,2,3,4,12,24 only
	1,2,3,4,6,12,24 only
	1,2,3,4,8,12,24 only
	1,2,3,4,6,8,12,24 only

3 What are the possible orders of quotient groups of the symmetric group of degree four?

	1,2,6,24 only
	1,3,8,24 only
	1,4,12,24 only
	1,2,3,4,12,24 only
	1,2,3,4,6,12,24 only
	1,2,3,4,8,12,24 only
	1,2,3,4,6,8,12,24 only

4 For which of the following orders of subgroups do there exist two distinct conjugacy classes of subgroups in the symmetric group of that order for which the subgroups are isomorphic as abstract groups?

	2 only
	3 only
	4 only
	6 only
	2 and 3 only
	3 and 4 only
	2 and 4 only
	3 and 6 only