Point-transitive and 1-rotational unitals of order 5
Ivan Hetman, Taras Banakh, Alex Ravsky
TL;DR
The paper advances the classification of unitals of order $5$ by enumerating both point-transitive and $1$-rotational unitals, realized as Steiner systems $S(2,6,126)$. It leverages algebraic construction alongside GAP-derived group data and Cayley-table based generation of blocks via a difference-family framework, augmented by the hyperbolic frequency fingerprint to differentiate designs. The results yield a large catalog of non-isomorphic unitals across selected groups of orders $125$ and $126$, including explicit fingerprint-difference-family data and block lists, while highlighting overlaps with known designs in the Desarguesian plane of order $25$. This work provides a computationally explicit framework for classifying symmetric unitals of order $5$, facilitating further structural and intersection investigations in finite geometry.
Abstract
In this paper we introduce enumeration of unitals of order $5$, which are also Steiner systems $S(2,6,126)$, where automorphism group acts transitively and effectively on points or fixes one point.