On Bravais colourings of cyclotomic integers with class number one

Given a Bravais colouring of planar modules ${\text{M}}_n:=\mathbb{Z}[{\xi }_n]$, where ${\xi }_n$ is a primitive $n$th root of unity, two important colour groups arise: the colour symmetry group $H$, which permutes the colours of a given colouring of ${\text{M}}_n$, and the colour preserving group $K$, a normal subgroup of $H$ that fixes the colours. This paper gives a complete characterisation of $H$ and $K$ for all $\mathrm{\ell }$-colourings of ${\text{M}}_n$ for values of $n$ for which ${\text{M}}_n$ has class number one.