  Given a Bravais colouring of planar modules $\M_n:=\Z[\xi_n]$, 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
$\M_n$, 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 $\ell$-colourings of $\M_n$ for values of $n$ for which $\M_n$ has
class number one.
