Example of 3-dimensional nonabelian Lie group

Question

Could you please give an example of a Lie group diffeomorphic to $S^1\times \mathbb{R}^2$? Okay, $S^1\times \mathbb{R}^2$ suits us. What about nonabelian one?

Answer 1

Take a nontrivial semidirect product $\mathbb{R}^2 \rtimes S^1$, e.g. with $S^1$ acting by rotations in the usual way. This is the group of orientation-preserving isometries of the Euclidean plane.