Let $n\in\mathbb{N}$. For each $A\in \mathcal{M}_{n\times n}(\mathbb{R})$ and $b\in\mathbb{R}^{n}$, define $T_{A,b}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ the affine transformation
$$T_{A,b}(x)=Ax+b$$
Set $G=\{T_{A,b}: det(A)\neq 0\}$. It is easy to see that $G$ is a group under the composition of functions. Is this group a solvable and not nilpotent group?
I'm reading the book "Affine Maps, Euclidean Motions and Quadrics" but there is no mention of that on it.
If $n>1$, this group is not solvable since it contains $Sl(n,\mathbb{R})=\{ A\in Gl(n,\mathbb{R}), det(A)=1\}$ which is semi-simple.