# Schur cover of alternating group:A6

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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