What are all possible subdirect products of two quasisimple groups?

Question

Let $H$ be a proper subdirect product of two quasisimple groups $G_1$ and $G_2$. We know from the definition of quasisimple group that all the normal subgroups of $G_i$ live in $Z(G_i)$, the center of $G_i$. Then applying Goursat's lemma, we know that there exists $Z_i \leq Z(G_i)$, such that $G_1/Z_1 \cong G_2/Z_2$, and $H$ satisfies $\require{AMScd}$ \begin{CD} 1 @>>> & Z_2 @>>> & H @>>> & G_1 @>>> & 1, \end{CD} \begin{CD} 1 @>>> & Z_1 @>>> & H @>>> & G_2 @>>> & 1, \end{CD} and \begin{CD} 1 @>>> & Z_1 \times Z_2 @>>> & H @>>> & G_1/Z_1 \cong G_2/Z_2 @>>> & 1. \end{CD} My question is, since we require $G_1$ and $G_2$ to be quasisimple groups, what can we say about $H$ more than Goursat's lemma? For example, is it true that $H \cong G_1 \times Z_2$ or $H \cong G_2 \times Z_1$? In the special case where there exists a surjective homomorphism $\phi: G_1 \rightarrow G_2$, I could show that $H \cong G_1 \times Z_2$.

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Edit on Aug. 16: As pointed out by @DerekHolt, it is not necessarily true that $H \cong G_1 \times Z_2$ or $H \cong G_2 \times Z_1$. Now I wonder whether this weaker version is true: $H \cong G \times Z$, where $G$ is a perfect central extension of both $G_1$ and $G_2$, and $Z$ is a simutaneous subgroup of both $Z_1$ and $Z_2$. Or even weaker: is $[H, H]$ a perfect group?

Answer 1

The answer to your question "is $H \cong G_1 \times Z_2$ or $Z_1 \times G_2$" is no (i.e. not necessarily).

As an example, the Schur Multiplier of $A_6$ is cyclic of order $6$, so we take $G_1$ and $G_2$ to be $2$-fold and $3$-fold coverings of $A_6$, respectively (i.e. $2.A_6$ and $3.A_6$ in ATLAS notation).

Now if we take $Z_1 = Z(G_1)$ (of order $2$) and $Z_2 = Z(G_2)$ (of order $3$), then $H = 6.A_6$ is the full covering group of $A_6$.