# What's the Lie group generated only by dilation and Poincaré symmetry?

Given space $\mathbb{R}^{1,d-1}$($d\ge3$), the total conformal group is $SO(d,2)$ generated by $1$-dilation, $d$-translation, $d$-special conformal, $d(d-1)/2$-Lorentz transformation.

But we know generators of Poincare symmetry and dilation are closed, so form a Lie subalgebra of Lie algebra $so(d,2)$. So what's the Lie group corresponding to Lie algebra generated only by Poincare symmetry and dilation? Is it a classical Lie group or (semi-)direct product of classical Lie group or others?

The only nontrivial commutator between dilation and Poincare symmetry is $$[D,P_\mu]=iP_\mu$$

This is sometimes called the Weyl Group $$W(1,d-1)\sim\mathbb R \ltimes\mathrm{ISO}(1,d-1)$$