Affine group, identification and multiplication law

Question

I have a question about the group of affine transformations of $\mathbb{R^2}$. Where by that I mean the following: $Aff(\mathbb{R^2})=\{AX + b\mid A \in GL_2(\mathbb{R^2}), b \in \mathbb{R^2}\}=\left\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} p \\ q \end{bmatrix} \right\}$ I think I can identify, topologically this group with the following cartesian product $\mathbb{R^{*4}}\times \mathbb{R^2} \simeq \mathbb{R^6}$ where $\mathbb{R^{*4}} $ stands for the euclidean four dimensional space without the origin. My question is: is there a simpler topological identification of that group such that I can induce a multiplication law that completely determine the group? In $\mathbb{R^6}$ i found the following product: $(a,b,c,d,p,q)*(a',b',c',d',p',q')=(aa'+ bc',ab'+bd',a'c+dc',cb'+dd',a'p+b'q+p',c'p+d'q+q')$ The problem specifically is the following: since I need to calculate the left invariant form, I was asking myself if I can define something easier than that. I hope my question was clear enough.