It is a subgroup of the affine group $Aff(n)$. They are linear isometries composed with translations.
We have $E(n)=\mathbb{R}^n\times O(n)$. Indeed is a semidirect product
$$ E(n)=\mathbb{R}^{n} \rtimes O\left(n \right) $$To see it, taking into account section semidirect product#Remarks, you can think of the map
$$ \phi: E(n)\longrightarrow O\left(n \right) $$that associates to any $g\in E(n)$ the map $g$ itself composed with the translation that sends $g(0)$ to 0.
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Author of the notes: Antonio J. Pan-Collantes
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