# Direct product of SL(2,5) 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 special linear group:SL(2,5) (order 120), which is also the double cover of alternating group:A5 (and also its Schur covering group).
2. The group special linear group:SL(2,7) (order 336), which is also the double cover of PSL(3,2) (and also its Schur covering group).

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 40320#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 40320 groups with same order order of direct product is product of orders, so the order is $120 \times 336 = 40320$

## 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 No
perfect group Yes

## GAP implementation

Description Functions used
DirectProduct(SL(2,5),SL(2,7)) DirectProduct, SL
PerfectGroup(40320,1) PerfectGroup