# Difference between revisions of "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 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