  Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and
such that every group of order $np$ has a normal subgroup of order $p$. (This
holds in particular for $p>n$.) We prove that left braces of size $np$ may be
obtained as a semidirect product of the unique left brace of size $p$ and a
left brace of size $n$. We give a method to determine all braces of size $np$
from the braces of size $n$ and certain classes of morphisms from the
multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it
we derive a formula giving the number of Hopf Galois structures of abelian type
$\mathrm{Z}_p \times E$ on a Galois extension of degree $np$ in terms of the
number of Hopf Galois structures of abelian type $E$ on a Galois extension of
degree $n$. For a prime number $p\geq 7$, we apply the obtained results to
describe all left braces of size $12p$ and determine the number of Hopf Galois
structures of abelian type on a Galois extension of degree $12p$.
