A census of small Schurian association schemes

TL;DR

This paper provides a comprehensive classification and analysis of small Schurian association schemes and transitive groups of degree up to 48, including character tables and 2-closures, with results available in a database.

Contribution

It offers the first complete classification of Schurian association schemes for degrees up to 48, including character tables and 2-closure computations, expanding understanding of these algebraic structures.

Findings

Classified all Schurian association schemes of order n for 2 ≤ n ≤ 48.

Computed character tables for each association scheme.

Determined the 2-closure of each transitive group within the same range.

Abstract

Using the classification of transitive groups of degree $n$, for $2 \leqslant n \leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 \leqslant n \leqslant 48$. The results of this classification are made available as a supplementary database.