What is the dimension of the Euclidean motion groups?

Question

The group of Euclidean motions of $\mathbb R^n$ is the semi-direct product $G=\mathbb R^n\rtimes K$ with $K=SO(n, \mathbb R^n)$. Elements of G are written as pairs $g=(x,k)$.

I would like to know what is the dimension of $G$ ? Thank you in advance

Answer 1

$SO(n)$ has dimension $\binom{n}{2}$ and so your $G$ would have dimension $n + \binom{n}{2}$.