Latin quandles of size $16p$

Abstract

In this paper we obtain the classification of latin quandles of size $16p$ where $p$ is an odd prime. In particular we have that such quandles are always subdirectly reducible but in some cases for small primes. Specifically, we have that latin quandles of size $16p$ are affine if $p\ne 1(\mathrm{mod}3)$ or $p\ne 3,5$. If $p=1(\mathrm{mod}3)$ there are $2$ subdirectly reducible non directly decomposable latin quandles of size $16p$ for every prime with $p=1(\mathrm{mod}3)$ and there are one subdirectly irreducible latin quandle of size $16p$ for $p=3,5$. We provide explicit constructions as coset quandles for all the quandles mentioned above.