# 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