Construction of All Gyrogroups of Orders at most 31

The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\star$ containing an identity element such that each element has an inverse. Furthermore, for each pair $(a,b)$ of elements of this structure there exists an automorphism $\text{gyr}a,b$ with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders $p,2p$ and $p^2$ are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.