The non-commuting, non-generating graph of a non-simple group

Let $G$ be a (finite or infinite) group such that $G\mathrm{/}Z(G)$ is not simple. The non-commuting, non-generating graph $\mathrm{\Xi }(G)$ of $G$ has vertex set $G\setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y]\mathrm{\ne }1$ and $⟨x,y⟩\mathrm{\ne }G$. We investigate the relationship between the structure of $G$ and the connectedness and diameter of $\mathrm{\Xi }(G)$. In particular, we prove that the graph either: (i) is connected with diameter at most $4$; (ii) consists of isolated vertices and a connected component of diameter at most $4$; or (iii) is the union of two connected components of diameter $2$. We also describe in detail the finite groups with graphs of type (iii). In the companion paper arXiv:2212.01616, we consider the case where $G\mathrm{/}Z(G)$ is finite and simple.