Supergroups of alternating group:A6

This article gives specific information, namely, supergroups, about a particular group, namely: alternating group:A6.
View supergroups of particular groups | View other specific information about alternating group:A6

This article discusses some of the groups that admit the alternating group of degree six as a subgroup, quotient group, or subquotient.

Note that unlike the discussion of the subgroup structure of alternating group:A6, this discussion is necessarily not comprehensive, because there are infinitely many groups containing the alternating group of degree four.

Subgroups and quotients: essential minimalist examples

Subgroups: making some or all the outer automorphisms inner

All the groups listed here are almost simple groups, because alternating group:A6 is a simple non-abelian group.

The outer automorphism group of alternating group:A6 is a Klein four-group. In particular, it has order 4. By the fourth isomorphism theorem, subgroups of the automorphism group containing the inner automorphism group correspond to subgroups of the outer automorphism group, which is the quotient group of the automorphism group by the inner automorphism group.

Since alternating group:A6 is a centerless group, it is identified naturally with its inner automorphism group, so each of the subgroups of the automorphism group containing the inner automorphism group is also a group containing $A_{6}$ as a self-centralizing normal subgroup. The whole automorphism group contains $A_{6}$ as a NSCFN-subgroup.

Below is the complete list of these groups.:

Group containing $A_{6}=\operatorname {Inn} (A_{6})$ and contained in $\operatorname {Aut} (A_{6})$ (this is an almost simple group)	Corresponding subgroup of $\operatorname {Out} (A_{6})$ viewed as Klein four-group	Order of group	Order of corresponding subgroup of $\operatorname {Out} (A_{6})$ = index of $A_{6}$ in group = order of group/360	Second part of GAP ID of big group (GAP ID is (order,2nd part))
alternating group:A6	trivial subgroup	360	1	118
symmetric group:S6	one of the three copies of Z2 in V4	720	2	763
projective general linear group:PGL(2,9)	one of the three copies of Z2 in V4	720	2	764
Mathieu group:M10	one of the three copies of Z2 in V4	720	2	765
automorphism group of alternating group:A6	whole group	1440	4	5841

Quotients: Stem extensions and Schur covering groups

All the groups listed here are quasisimple groups, because alternating group:A6 is a simple non-abelian group.

Further information: Group cohomology of alternating group:A6#Schur multiplier, group cohomology of alternating groups

The Schur multiplier of alternating group:A6 is cyclic group:Z6, i.e., the group $\mathbb {Z} /6\mathbb {Z}$. For each of the possible quotient groups of $\mathbb {Z} /6\mathbb {Z}$, there is a unique stem extension with that as base normal subgroup and alternating group:A6 as quotient. The stem extension for the whole Schur multiplier is the unique Schur covering group, also called the universal central extension.

Note that uniqueness follows from $A_{6}$ being a perfect group.

The list is below:

Group at base of stem extension	Order	Corresponding stem extension group (this is a quasisimple group)	Order	Second part of GAP ID (GAP ID is (order,second part))
trivial group	1	alternating group:A6	360	118
cyclic group:Z2	2	special linear group:SL(2,9), also denoted $2\cdot A_{6}$ to indicate that it is a double cover of alternating group	720	409
cyclic group:Z3	3	triple cover of alternating group:A6	1080	260
cyclic group:Z6	6	Schur cover of alternating group:A6	2160	(ID not available for this order)