View source for Central product of SL(2,5) and SL(2,7)

{{particular group}}

==Definition==

This group is defined as the [[central product]] of [[special linear group:SL(2,5)]] (order 120) and [[special linear group:SL(2,7)]] (order 336) where we perform the unique identification of the centers of these two groups with each other.

==Arithmetic functions==

{{compare and contrast arithmetic functions|order = 20160}}

{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation
|-
| {{arithmetic function value order|20160}} || Using the [[product formula]], we get <math>|SL(2,5)||SL(2,7)|/2</math> where the "2" comes because that's the order of the subgroup identified. This gives <math>120 * 336/2 = 20160</math>.
|}

==Group properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation
|-
| [[dissatisfies property::abelian group]] || No ||
|-
| [[dissatisfies property::nilpotent group]] || No ||
|-
| [[dissatisfies property::solvable group]] || No ||
|-
| [[dissatisfies property::simple group]], [[dissatisfies property::simple non-abelian group]] || No ||
|-
| [[dissatisfies property::quasisimple group]]|| No ||
|-
| [[satisfies property::directly indecomposable group]] || Yes ||
|-
| [[satisfies property::perfect group]] || Yes||
|}

==GAP implementation==

{| class="sortable" border="1"
! Description !! Functions used
|-
| <tt>PerfectGroup(20160,3)</tt> || [[GAP:PerfectGroup|PerfectGroup]]
|}