Symmetric group:S7

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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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

Upto 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


GAP implementation

Description Functions used
SymmetricGroup(7) SymmetricGroup