This is a classification of coherent configurations with at most 13 vertices.
First, Megumi Shiratsuchi classified coherent configurations at most 11 vertices in her master thesis in 1997. It was not complete. One of the reasons of the incompleteness is that classification of association schemes was not complete then.
In 2009, Asumi Nagatomo and Junichi Shigezumi made a correction of her result, and classified coherent configurations with 12 and 13 vertices. Most of them are published in the master thesis of Asumi Nagatomo. Most of the classification was done by hand, and we also use `Mathematica' partially.
The following table is the total numbers of coherent configurations:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
1 | 2 | 4 | 10 | 15 | 38 | 57 | 143 | 227 | 491 | 766 | 1821 | 2771 |
Every coherent configuration with at most 13 vertices is Schurian. (i.e. each configuration corresponds to some permutation group.)
The following datas include lists of relation matirices of coherent configurations with at most 13 vertices, which are not association schemes and have no fiver of order 1:
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Additional remark
The above table and this database of coherent configuration are not incomplete.
Prof. Matan Ziv-Av told us some example of coherent configuration which are not included in our list.
However, we could not check and correct our list yet.
Also, there is another project by Sven Reichard:
http://conferences2.imfm.si/conferenceDisplay.py/getPic?amp;confId=12&picId=20