Finite subgroups of $SL_2(\mathbb{C})$ arising as a semi-direct product

Question

I am trying to find all finite subgroups of $SL_2(\mathbb{C})$ that arise as a semi-direct product of two finite (non-trivial) subgroups of $SL_2(\mathbb{C})$. Is there any such characterization? Any hint/reference will be most welcome.

Answer 1

Reducible finite subgroups $G$ of ${\rm SL}(2,\mathbb{C})$ are Abelian (and even cyclic), so these can be dealt with easily.

Every finite cyclic group can occur, and only the finite cyclic $p$-groups are not direct products of two non-trivial subgroups.

Irreducible, but imprimitive, subgroups $G$ of ${\rm SL}(2,\mathbb{C})$ have an cyclic normal subgroup $A$ of index $2$, using unimodularity. Furthermore, (again by unimodularity), $A$ contains every element of order $2$ in $G$, so $A$ is not complemented.

If $A$ is not a $2$-group, then $G$ has a (cyclic) normal subgroup $H$ of odd order which is complemented by a Sylow $2$-subgroup $S$, so $G$ is a semidirect product $HS$.

If $A$ is a $2$-group, then $G$ has only one element of order $2$, and $G$ is a generalized quaternion group, so is not a semidirect product.

This leaves irreducible primitive groups. Such a group $G$ has all Abelian normal subgroups central, so of order $2$ by unimodularity and Schur's Lemma. If $G$ is solvable, then $F(G)$ is a $2$-group, (but $G$ is not a $2$-group by primitivity), so $F(G)$ is quaternion of order $8$ ( any larger generalized quaternion $2$-group has automorphism group a $2$-groups, so can't be $F(G)$ as $G$ is not a $2$-group). Then unimodularity forces either $G \cong {\rm SL}(2,3)$ or $G$ a binary octahedral group ( with generalized quaternion Sylow $2$-subgroup). The first possibility is a semidirect product, the second is not. Unimodularity excludes $G \cong {\rm GL}(2,3)$.

If $G$ is not solvable, then a minimal normal subgroup of $G/Z(G)$ must be simple. I won't go through all arguments, but the only possibility that occurs is $G \cong {\rm SL}(2,5)$, as is well known, and this is not a semidirect product. Note that unimodularity (among other things) excludes $G \cong {\rm GL}(2,5).$