Schur cover of alternating group:A6

From Groupprops
Jump to: navigation, search
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

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