What is the smallest split-simple non-simple group other than generalized quaternion groups and cyclic groups?

Question

The quaternion group $Q_8$ is split-simple, i.e. it cannot be written as an internal semidirect product of proper subgroups. In fact all generalized quaternion groups are split-simple, as are all simple groups and many cyclic groups.

My question is, what is the smallest group which is split-simple but which is neither a generalized quaternion group, a simple group, or a cyclic group?

Answer 1

The smallest examples you are looking for have order 32. A computation with GAP/Magma finds the following examples:

SmallGroup(32,8), a nonsplit extension $C_4.D_8$.

SmallGroup(32,15), also a nonsplit extension $C_4.D_8$.

SmallGroup(32,32), a nonsplit extension $C_4.Q_8$.

There are many more examples of larger orders.

An infinite family of examples is given by covering groups of simple alternating groups, i.e. the central extensions $2.A_n$.