# Central product of SL(2,5) and SL(2,7)

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## 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

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 20160#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 20160 groups with same order Using the product formula, we get $|SL(2,5)||SL(2,7)|/2$ where the "2" comes because that's the order of the subgroup identified. This gives $120 * 336/2 = 20160$.

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group No
quasisimple group No
directly indecomposable group Yes
perfect group Yes

## GAP implementation

Description Functions used
PerfectGroup(20160,3) PerfectGroup