Schur cover of alternating group:A6

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

This group, termed the Schur cover of alternating group:A6 and sometimes denoted $6\cdot A_{6}$, is defined in the following equivalent ways:

1. It is the unique quasisimple group with center cyclic group:Z6 and quotient group alternating group:A6.
2. It is the Schur covering group of alternating group:A6.
3. It is the Schur covering group of special linear group:SL(2,9).

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2160#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	2160	groups with same order	As the Schur covering group of $A_{6}$: order of Schur multiplier of $A_{6}$ (which is 6) times order of $A_{6}$ (which is $6!/2=360$)

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group, simple non-abelian group	No	nontrivial center of order six
almost simple group	No
quasisimple group	Yes
almost quasisimple group	Yes
perfect group	Yes

GAP implementation

Description	Functions used
SchurCover(AlternatingGroup(6))	SchurCover, AlternatingGroup
SchurCover(SL(2,9))	SchurCover, SL
PerfectGroup(2160) or equivalently PerfectGroup(2160,1)	PerfectGroup