Triple cover of alternating group:A6

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Definition

This group, termed the triple cover of alternating group:A6, and denoted $3\cdot A_{6}$ , is defined in the following equivalent ways:

It is the unique quasisimple group whose center is isomorphic to cyclic group:Z3 and inner automorphism group is alternating group:A6.
It is the quotient of the Schur cover of alternating group:A6 by a subgroup of order two inside its center (which is isomorphic to cyclic group:Z6).

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 1080#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	1080	groups with same order	As $3\cdot A_{6}$ : $3\|A_{6}\|=3(6!)/2=3(360)=1080$
exponent of a group	60	groups with same order and exponent of a group \| groups with same exponent of a group

Arithmetic functions of an element-counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes		groups with same order and number of conjugacy classes \| groups with same number of conjugacy classes

Group properties

Property	Satisfied?	Explanation
simple group, simple non-abelian group	No	center is a proper nontrivial normal subgroup
quasisimple group	Yes
solvable group	No
abelian group	No	(via solvable)
nilpotent group	No	(via solvable)

GAP implementation

Group ID

This finite group has order 1080 and has ID 260 among the groups of order 1080 in GAP's SmallGroup library. For context, there are groups of order 1080. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(1080,260)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(1080,260);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [1080,260]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description	Functions used
`PerfectGroup(1080)` or equivalently `PerfectGroup(1080,1)`	PerfectGroup