Symmetric group:S8

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
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 $8$ . The set is typically taken to be $\{1,2,3,4,5,6,7,8\}$ .

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

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	40320	groups with same order	The order is $8!=8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$
exponent of a group	840	groups with same order and exponent of a group \| groups with same exponent of a group	The exponent is the least common multiple of $1,2,3,4,5,6,7,8$
Frattini length	1	groups with same order and Frattini length \| groups with same Frattini length

Elements

Further information: element structure of symmetric group:S8

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 + 1	all points fixed	$()$ -- the identity element	1	${\frac {8!}{(1)^{8}(8!)}}$	even;no	1
2 + 1 + 1 + 1 + 1 + 1 + 1	transposition, six fixed points	$(1,2)$	28	${\frac {8!}{(2)(1)^{6}(6!)}}$ , also ${\binom {8}{2}}$	odd	2
3 + 1 + 1 + 1 + 1 + 1	one 3-cycle, five fixed points	$(1,2,3)$	112	${\frac {8!}{(3)(1)^{5}(5!)}}$	even;no	3
4 + 1 + 1 + 1 + 1	one 4-cycle, four fixed points	$(1,2,3,4)$	420	${\frac {8!}{(4)(1)^{4}(4!)}}$	odd	4
2 + 2 + 1 + 1 + 1 + 1	two transpositions, four fixed points	$(1,2)(3,4)$	210	${\frac {8!}{(2)^{2}(2!)(1)^{4}(4!)}}$	even;no	2
5 + 1 + 1 + 1	one 5-cycle, three fixed points	$(1,2,3,4,5)$	1344	${\frac {8!}{(5)(1)^{3}(3!)}}$	even;no	5
3 + 2 + 1 + 1 + 1	one 3-cycle, one transposition, three fixed points	$(1,2,3)(4,5)$	1120	${\frac {8!}{(3)(2)(1)^{3}(3!)}}$	odd	6
6 + 1 + 1	one 6-cycle, two fixed points	$(1,2,3,4,5,6)$	3360	${\frac {8!}{(6)(1)^{2}(2!}}$	odd	6
4 + 2 + 1 + 1	one 4-cycle, one 2-cycle, two fixed points	$(1,2,3,4)(5,6)$	2520	${\frac {8!}{(4)(2)(1)^{2}(2!)}}$	even;no	4
2 + 2 + 2 + 1 + 1	three 2-cycles, two fixed points	$(1,2)(3,4)(5,6)$	420	${\frac {8!}{(2)^{3}(3!)(1)^{2}(2!)}}$	odd	2
3 + 3 + 1 + 1	two 3-cycles, two fixed points	$(1,2,3)(4,5,6)$	1120	${\frac {8!}{(3)^{2}(2!)(1)^{2}(2!)}}$	even;no	3
7 + 1	one 7-cycle, one fixed point	$(1,2,3,4,5,6,7)$	5760	${\frac {8!}{(7)(1)}}$	even;yes;no	7
3 + 2 + 2 + 1	one 3-cycle, two transpositions, one fixed point	$(1,2,3)(4,5)(6,7)$	1680	${\frac {8!}{(3)(2)^{2}(2!)(1)}}$	even;no	6
4 + 3 + 1	one 4-cycle, one 3-cycle, one fixed point	$(1,2,3,4)(5,6,7)$	3360	${\frac {8!}{(4)(3)(1)}}$	odd	12
5 + 2 + 1	one 5-cycle, one 2-cycle, one fixed point	$(1,2,3,4,5)(6,7)$	4032	${\frac {8!}{(5)(2)(1)}}$	odd	10
2 + 2 + 2 + 2	four 2-cycles	$(1,2)(3,4)(5,6)(7,8)$	105	${\frac {8!}{(2)^{4}(4!)}}$	even;no	2
4 + 2 + 2	one 4-cycle, two 2-cycles	$(1,2,3,4)(5,6)(7,8)$	1260	${\frac {8!}{(4)(2)^{2}(2!)}}$	odd	4
3 + 3 + 2	two 3-cycles, one 2-cycle	$(1,2,3)(4,5,6)(7,8)$	1120	${\frac {8!}{(3)^{2}(2!)(2)}}$	odd	6
6 + 2	one 6-cycle, one 2-cycle	$(1,2,3,4,5,6)(7,8)$	3360	${\frac {8!}{(6)(2)}}$	even;no	6
5 + 3	one 5-cycle, one 3-cycle	$(1,2,3,4,5)(6,7,8)$	2688	${\frac {8!}{(5)(3)}}$	even;yes;no	15
4 + 4	two 4-cycles	$(1,2,3,4)(5,6,7,8)$	1260	${\frac {8!}{(4)^{2}(2!)}}$	even;no	4
8	one 8-cycle	$(1,2,3,4,5,6,7,8)$	5040	${\frac {8!}{8}}$	odd	8