Symmetric group:S7

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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 $7$. 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	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"{{{" can not be assigned to a declared number type with value 3. | 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"{{{" can not be assigned to a declared number type with value 3. | groups with same Frattini length