Concise notation for the central extension defined by a cocycle

Question

Any extension of a group $G$ by the Abelian group $A$ is determined (up to isomorphism) by $\varphi$ and $f$, where

1. $\varphi:G\times A\to A$ is a group action of $G$ on $A$
2. $f:G\times G\to A $ is a cocycle.

(see this question)

• The special case when $f=0$ and $\varphi$ is trivial is the direct product of $A$ and $G$ and it is denoted by $A\times G$.

• The special case when $f=0$, but $\varphi$ is possible nontrivial is the semidirect product of $A$ and $G$ and it is denoted by $A\rtimes_\varphi G$

• The special case when $\varphi$ is trivial, but $f$ is an arbitrary cocycle is the central extension of $G$ by $A$ and I didn't find a concise notation for it (it should contain $A$, $G$ and $f$). Is there any? (if isn't there any, then what about $A\otimes_f G$ or $A\odot_f G$?)

(and perhaps the general case also deserves some concise notation, containing $A$, $G$, $\varphi$ and $f$)

Answer 1

I've found a notation in nLab (definition 3.10). According to this notation, the central extension of $G$ by $A$ defined by cocycle $f$ is $$G\times_f A$$