Does anything special happen when you replace the direct product in the definition for a wreath product with a central product? In group theory there is a special type of product called the wreath product and is defined as follows:
Let $A$ and $H$ be groups, with the group $H$ acting on the set $\Omega$. We define $K$ to be the direct product:
$$K = \prod_{\omega \in \Omega} A_{\omega}$$
the duplicates, $A_{\omega}$ of the group $A$ being indexed by elements of the set $\Omega$.
What happens when we change the direct product in this definition with a central product? Does that change anything? If the answer is yes, then does this type of product have a name?
 A: If $Z \le Z(A)$ then you can define the analogue of the wreath product in which the copies $Z_\omega$ of $Z$ in $A_\omega$ are amalgamated, and the result is the quotient of the ordinary wreath product by a central subgroup of the base group.
I am not aware that this has a name, but there is one context in which it arises naturally.
For simplicity, let's suppose $\Omega$ is finite with $|\Omega| = n$. Suppose that $A$ is a matrix group (i.e. a subgroup of a general linear group ${\rm GL}(d,K)$), and $V$ is the associated $KA$-module.Then you can define a tensor wreath product of $A$ by $H$ as a subgroup of ${\rm GL}(d^n,K)$, in which the module for the base group is the tensor power of $n$ copies of $V$.
When you do that, some of the scalar matrices in the direct product $A^n$ can  be equal to the identity. Specifically, that happens to the element $(\lambda_1 I_d,\lambda_2 I_d,\ldots,\lambda_n I_d) \in A^n$, when $\lambda_1\lambda_2 \cdots \lambda_n = 1$, and you end up with a quotient of the wreath product of the type in which the base group is a central product with amalgamted subgroup isomorphic to the scalar subgroup of $A$.
If you are familiar with Aschbacher's classification of matrix groups over finite fields, then tensor powered groups of this type form Class ${\mathcal C}_7$.
