# Direct product of A5 and SL(2,7)

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## Definition

This group is defined as the external direct product of the following two groups:

1. The group $A_5$, i.e., alternating group:A5 (order 60), which is also $PSL(2,4)$ and $PSL(2,5)$.
2. The group $SL(2,7)$, i.e., special linear group:SL(2,7) (order 336), which is also the double cover of PSL(3,2).

## 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 order of direct product is product of orders, so the order is $60 \times 336 = 20160$

## Group properties

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

## GAP implementation

Description Functions used
DirectProduct(AlternatingGroup(5),SL(2,7)) DirectProduct, AlternatingGroup, SL
PerfectGroup(20160,1) PerfectGroup