# Determining skew left braces of size np

## TL;DR

This paper characterizes skew left braces of size np where p is an odd prime and n meets specific conditions, providing a classification, an algorithm for enumeration, and confirming a conjecture for size 12p.

## Contribution

It introduces a new structural decomposition for skew left braces of size np and develops an algorithm to enumerate all such braces, confirming a conjecture for size 12p.

## Key findings

- Classified skew left braces of size np under certain conditions.
- Developed an algorithm to enumerate all skew left braces of size np.
- Proved the conjecture for skew left braces of size 12p with p ≥ 7.

## Abstract

We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an integer $n$ satisfying $p\nmid n$, $p\nmid|\mathrm{Aut}(E)|$ for every group $E$ of order $n$ and such that each group of order $np$ has a unique $p$-Sylow subgroup. Under these conditions, we prove that any skew left brace of size $np$ is either a twofold semidirect product of the trivial brace of size $p$ and a skew left brace of size $n$ or a companion skew left brace of that one. We develop an algorithm to obtain all skew left braces of size $np$ from the skew left braces of size $n$ and provide a formula to count them. We use this result to describe all skew left braces of size $12p$ for $p\geq 7$, which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.