Supergroups of special linear group:SL(2,5)

This article discusses some of the groups that admit special linear group:SL(2,5) as a subgroup, quotient group, or subquotient.

Note that unlike the discussion of subgroup structure of special linear group:SL(2,5), the discussion is necessarily not comprehensive, because there are infinitely many groups containing any given group.

Subgroups: making all automorphisms inner
The automorphism group of SL(2,5) is isomorphic to symmetric group:S5, which can be more explicitly thought of as PGL(2,5) in this context. Explicitly, the elements of $$GL(2,5)$$ act as automorphisms of $$SL(2,5)$$ by conjugation, and the kernel of this action is the center, so this induces a faithful action of $$PGL(2,5)$$ on $$SL(2,5)$$ and an embedding of $$PGL(2,5)$$ inside the automorphism group of $$SL(2,5)$$. In this case, that embedding is surjective, i.e., $$SL(2,5)$$ has no further automorphisms.

There are a number of ways we can find a group $$G$$ containing a subgroup $$H$$ isomorphic to $$SL(2,5)$$ as a normal subgroup so that the induced action by conjugation yields a surjective homomorphism $$G/C_G(H) \to \operatorname{Aut}(H)$$ (in other words, $$G$$ contains a normal fully normalized subgroup isomorphic to $$SL(2,5)$$).

The minimal examples are those where $$H$$ is a self-centralizing subgroup of $$G$$ (i.e., $$C_G(H) = Z(H)$$), i.e., is a NSCFN-subgroup of $$G$$. In this case, $$G$$ has normal subgroup $$Z(H)$$ of order two, with quotient group $$\operatorname{Aut}(H)$$ of order 120. In particular, $$G$$ is a double cover of $$\operatorname{Aut}(H) \cong S_5$$. In particular, $$G$$ must have order 240.

The upshot is that the classification of groups that contain SL(2,5) as a NSCFN-subgroup is exactly the same as the classification of double covers of the automorphism group, which is isomorphic to symmetric group:S5. There are two such groups, both of order 240:


 * Double cover of symmetric group:S5 of plus type
 * Double cover of symmetric group:S5 of minus type

For more, see group cohomology of symmetric groups, double cover of symmetric group, group cohomology of symmetric group:S5, and supergroups of symmetric group:S5.

In addition, there are many non-minimal examples. For instance, general linear group:GL(2,5) is a group of order 480 containing SL(2,5) as a normal fully normalized subgroup of index four.

Quotients: Schur covering groups
The group special linear group:SL(2,5) is a Schur-trivial group, i.e., its Schur multiplier is a trivial group. Further, the group is a perfect group. Thus, its Schur covering group is itself.