Symmetric group:S7

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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