Supergroups of alternating group:A4
This article gives specific information, namely, supergroups, about a particular group, namely: alternating group:A4.
View supergroups of particular groups | View other specific information about alternating group:A4
Note that unlike the discussion of the subgroup structure of alternating group:A4, 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 all the automorphisms inner
The outer automorphism group of alternating group:A4 is cyclic group:Z2 and the automorphism group is symmetric group:S4. Since is centerless, it equals its inner automorphism group and hence embeds as a subgroup of index two inside symmetric group:S4.
Quotients: Schur covering groups
The Schur multiplier of alternating group:A4 is cyclic group:Z2. There is a unique corresponding Schur covering group, namely the group special linear group:SL(2,3), where the center of special linear group:SL(2,3) is isomorphic to the Schur multiplier cyclic group:Z2 and the quotient is alternating group:A4.
The Schur covering group is also denoted to indicate that it is a double cover of alternating group.