I've encounter a group with properties that are very familiar, but I cannot say what group is it.


Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables can be represented as 
$$ M =
\begin{pmatrix}
1 & 0
\\
\vec{a} & {R}
\end{pmatrix},$$
such that
$$
\begin{pmatrix}
t'
\\
\vec{x}'
\end{pmatrix} = 
\begin{pmatrix}
1 & 0
\\
\vec{a} & {R}
\end{pmatrix} 
\begin{pmatrix}
t
\\
\vec{x}
\end{pmatrix} = 
\begin{pmatrix}
t
\\
R\vec{x} + \vec{a}
\end{pmatrix}.
$$




---


The group I mentioned is composed by elements which are like the transpose of the above $M$, i.e.
$$
\begin{pmatrix}
t'
\\
\vec{x}'
\end{pmatrix} = 
\begin{pmatrix}
1 & \vec{a}
\\
0 & {R}
\end{pmatrix} 
\begin{pmatrix}
t
\\
\vec{x}
\end{pmatrix} = 
\begin{pmatrix}
t + \vec{a} \cdot \vec{x}
\\
R\vec{x} 
\end{pmatrix}.
$$
If $d = \dim(\vec{x})$, the above is a $d$-parametric affine group of dimension one! but we still have the possible mixing on the $x$ variables (i.e. $R$).


**Question:** 


Does this group have a name? Is it a semi-direct product of other groups?



Identification problem: Does this group have a name?
The group is a (homogeneous) Carroll group: see eq. 2.7 in <https://arxiv.org/abs/1405.2264>


