Direct product of A5 and SL(2,7)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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