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A loop is a quasigroup with the identity. I have to disclose that loop-theory is something outside my expertise.

I have four loops arising from octonionic elements of unitary norm that have order 16, 24, 48, 240 respectively. If needed I can provide more details on how elements behave. Is there some reference where low order loops are classified or treated as finite groups?

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    $\begingroup$ I don't know anything about loops, but I just googled around for a bit, and the Wikipedia article on Moufang loops mentions that the unit norm integral octonions form a Moufang loop with 240 elements, and the basis octonions form a Moufang loop with 16 elements. These seem to be on your list, so maybe that article is a starting point. (It also mentions a reference "The Moufang loops of order less than 64", which might be what you're looking for) $\endgroup$
    – Achim Krause
    Apr 26, 2023 at 7:53
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    $\begingroup$ To add to Achim's comment, you need to be careful with the definition of integer octonions. There's a classic error known as "Kirmse's mistake" (poor guy!) to correct which, you need to choose one of the seven pure imaginary coordinates and lose symmetry. See for example the second page of Rob Wilson's "Octonions and the Leech Lattice" in J Algebra 2009. With the correct definition, the 240 unit octonions form a copy of the root system in the $E_8$ lattice. They don't form a group, but rather a Moufang loop. $\endgroup$
    – Dave Benson
    Apr 26, 2023 at 9:07
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    $\begingroup$ I removed the tag "loop groups" which refers to something completely different. $\endgroup$
    – Vladimir Dotsenko
    Apr 26, 2023 at 11:19
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    $\begingroup$ There is a Loops library for GAP. See gap-system.org/Packages/loops.html $\endgroup$
    – Sean Eberhard
    Apr 26, 2023 at 13:13
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    $\begingroup$ @DaveBenson Thank you very much for your reference! It was very useful indeed $\endgroup$
    – Dac0
    Apr 26, 2023 at 13:25

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