Symmetric group:S7

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Definition

This group is a finite group defined as the symmetric group on a set of size $7$. The set is typically taken to be $\{1,2,3,4,5,6,7\}$.

In particular, it is a symmetric group on finite set as well as a symmetric group of prime degree.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	5040	groups with same order	The order is $7!=7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$
exponent of a group	420	groups with same order and exponent of a group | groups with same exponent of a group	The exponent is the least common multiple of $1,2,3,4,5,6,7$
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length

Elements

Further information: element structure of symmetric group:S7

Up to conjugacy

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	seven fixed points	$()$ -- the identity element	1	${\frac {7!}{(1)^{7}(7!)}}$	even; no	1
2 + 1 + 1 + 1 + 1 + 1	transposition, five fixed points	$(1,2)$	21	${\frac {7!}{(2)(1)^{5}(5!)}}$, also ${\binom {7}{2}}$ in this case	odd	2
3 + 1 + 1 + 1 + 1	one 3-cycle, four fixed points	$(1,2,3)$	70	${\frac {7!}{(3)(1)^{4}(4!)}}$	even; no	3
4 + 1 + 1 + 1	one 4-cycle, three fixed points	$(1,2,3,4)$	210	${\frac {7!}{(4)(1)^{3}(3!)}}$	odd	4
2 + 2 + 1 + 1 + 1	two 2-cycles, three fixed points	$(1,2)(3,4)$	105	${\frac {7!}{(2)^{2}(2!)(1)^{3}(3!)}}$	even;no	2
5 + 1 + 1	one 5-cycle, two fixed points	$(1,2,3,4,5)$	504	${\frac {7!}{(5)(1)^{2}(2!)}}$	even; no	5
3 + 2 + 1 + 1	one 3-cycle, one 2-cycle, two fixed points	$(1,2,3)(4,5)$	420	${\frac {7!}{(3)(2)(1)^{2}(2!)}}$	odd	6
6 + 1	one 6-cycle, one fixed point	$(1,2,3,4,5,6)$	840	${\frac {7!}{(6)(1)}}$	odd	6
4 + 2 + 1	one 4-cycle, one 2-cycle, one fixed point	$(1,2,3,4)(5,6)$	630	${\frac {7!}{(4)(2)(1)}}$	even;no	4
2 + 2 + 2 + 1	three 2-cycles, one fixed point	$(1,2)(3,4)(5,6)$	105	${\frac {7!}{(2)^{3}(3!)(1)}}$	odd	2
3 + 3 + 1	two 3-cycles, one fixed point	$(1,2,3)(4,5,6)$	280	${\frac {7!}{(3)^{2}(2!)(1)}}$	even;no	3
3 + 2 + 2	one 3-cycle, two transpositions	$(1,2,3)(4,5)(6,7)$	210	${\frac {7!}{(3)(2)^{2}(2!)}}$	even;no	6
5 + 2	one 5-cycle, one transposition	$(1,2,3,4,5)(6,7)$	504	${\frac {7!}{(5)(2)}}$	odd	10
4 + 3	one 4-cycle, one 3-cycle	$(1,2,3,4)(5,6,7)$	420	${\frac {7!}{(4)(3)}}$	odd	12
7	one 7-cycle	$(1,2,3,4,5,6,7)$	720	${\frac {7!}{7}}$	even;yes;no	7

Subgroups

Further information: subgroup structure of symmetric group:S7

Quick summary

Item	Value
Number of subgroups	11300 Compared with $S_{n},n=1,2,\dots$: 1,2,6,30,156,1455,11300,151221
Number of conjugacy classes of subgroups	96 Compared with $S_{n},n=1,2,\dots$: 1,2,4,11,19,56,96,296,554,1593,...
Number of automorphism classes of subgroups	96 Compared with $S_{n},n=1,2,\dots$: 1,2,4,11,19,37,96,296,554,1593,...
Isomorphism classes of Sylow subgroups	2-Sylow: direct product of D8 and Z2 (order 16) 3-Sylow: elementary abelian group:E9 (order 9) 5-Sylow: cyclic group:Z5 (order 5) 7-Sylow: cyclic group:Z7 (order 7)
Hall subgroups	Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{2,3\}$-Hall subgroups (of order 144) and $\{2,3,5\}$-Hall subgroups (of order 720), the latter being S6 in S7. Note that the $\{2,3\}$-Hall subgroups are not contained in $\{2,3,5\}$-Hall subgroups.
maximal subgroups	maximal subgroups have orders 42, 144, 240, 720, 2520
normal subgroups	the whole group, trivial subgroup, and alternating group:A7 (embedded as A7 in S7)
subgroups that are simple non-abelian groups	alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size $2^{2}+2+1=7$), alternating group:A6 (order 360), alternating group:A7 (order 2520)

Linear representation theory

Further information: linear representation theory of symmetric group:S7

Summary

Item	Value
degrees of irreducible representations over a splitting field	1,1,6,6,14,14,14,14,15,15,20,21,21,35,35 maximum: 35, lcm: 420, number: 15, sum of squares: 5040
Schur index values of irreducible representations over a splitting field	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)
smallest ring of realization (characteristic zero)	$\mathbb {Z}$ -- ring of integers
smallest field of realization (characteristic zero)	$\mathbb {Q}$ -- field of rational numbers
condition for a field to be a splitting field	Any field of characteristic not 2, 3, 5, or 7.
smallest size splitting field	field:F11, i.e., the field with 11 elements

GAP implementation

Description	Functions used
SymmetricGroup(7)	SymmetricGroup