# Changes

## Symmetric group:S7

, 00:45, 30 April 2012
no edit summary
{{particular group}}
[[importance rank::3| ]]
==Definition==
This [[group]] is a [[finite group]] defined as the [[member of family::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 [[member of family::symmetric group on finite set]] as well as a [[member of family::symmetric group of prime degree]].
==Arithmetic functions==
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation|-| {{arithmetic function value order|5040}}|| The order is $7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$
|-
| {{arithmetic function value given order|exponent of a group|420|5040}} || The order exponent is the least common multiple of $1,2,3,4,5,6,7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$
|-
| {{arithmetic function value given order|exponent Frattini length|1|5040}}|||} ==Elements== {{further|[[element structure of symmetric group:S7]]}} ===Up to conjugacy=== {{#lst:element structure of symmetric group:S7|conjugacy class structure}} ==Subgroups== {{further|[[subgroup structure of a symmetric group:S7]]}} {{#lst:subgroup structure of symmetric group:S7|420summary}} The exponent is the least common multiple  ==Linear representation theory== {{further|[[linear representation theory of $symmetric group:S7]]}} ===Summary==={{#lst:linear representation theory of symmetric group:S7|summary}}==GAP implementation== {| class="sortable" border="1,2,3,4,5,6,7$"! Description !! Functions used
|-
| {{arithmetic function value given order<tt>SymmetricGroup(7)</tt> |Frattini length|1}}[[GAP:SymmetricGroup|SymmetricGroup]]
|}