TITLE: Understand $\bowtie$ symbol : Non-commuting action of both group
QUESTION [5 upvotes]: Note this is a part of physics paper arXiv-9909150. And please assume that i am not familiar with advanced algebraic knowledge(Group theory, abstract algebra, etc..). 
Excerpt from above paper, 

The proposed U-duality group is then generated by
\begin{align}
G(Z)=SL(2,Z) \bowtie SO(d,d;Z),
\label{orig}
\end{align}
where $\bowtie$ refers to the non-commuting 
action of both groups.

I want to understand $\bowtie$ symbol, explicitly.  They said $\bowtie$ refers to the non-commuting action of both groups, but i am not pretty sure about its meaning. 
First, is $G(Z)$ is kind of direct product of $SL(2,Z)$ and $SO(d,d;Z)$? 
I mean G(Z) satisfies the both properties of $SL(2,Z)$ and $SO(d,d;Z)$? 
Second, non-commuting action means, $SL(2,Z) \bowtie SO(d,d;Z)$ and $SO(d,d;Z) \bowtie SL(2,Z)$ are different?

REPLY [4 votes]: I once wrote out a Math.SE answer about these in general here. This product goes by many names: Zappa-Szep product, the general product, the knit product, etc. They are the "next obvious thing" to define once you have defined direct products ($H,K\lhd G$, with $H\cap K=1$, $HK=G$) and semidirect products ($H\lhd G$, $K<G$, with $H\cap K=1$, $HK=G$). For Zappa-Szep we require that $H, K<G$, with $H\cap K=1$, $HK=G$. There are no conditions on the normality of the subgroups.
As an intriguing motivation, they can be studied for many, many algebraic structures: see this fascinating paper of Matt Brin for details!