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Question 1:Is there a reference that lists all possible finite subgroups and their orders of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$ for $n=4$ or even higher $n$ over the real numbers?

I can only find incomplete lists on wikipedia, mathoverflow or other sources in the internet.

Question 2: Is there a computer algebra package that can list those finite subgroups up to some order? Maybe a package of GAP?

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  • $\begingroup$ I think I asked question 1 of Ed Swartz (Cornell) once, and he knew of a Ph.D thesis where the complete list was given. Unfortunately I forget the reference, but perhaps try contacting Ed. $\endgroup$
    – Ryan Budney
    Commented Mar 3, 2023 at 1:35
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    $\begingroup$ All positive integers appear as orders of cyclic subgroups of orthogonal groups, so perhaps Question 1 should be made more fine-grained. For $n=4$, you can make a list of finite order subgroups using the product decomposition of $Spin(4)$ and the $n=3$ case. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 3, 2023 at 1:53
  • $\begingroup$ @S.Carnahan You are right. I changed the question so that it asks for the groups now. $\endgroup$
    – Mare
    Commented Mar 3, 2023 at 11:21
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    $\begingroup$ Tangentially relevant: there's a discussion of subgroups of $SO(4)$ that act freely on $S^3$ in Scott's The geometries of 3-manifolds. I don't know if anyone has written The geometries of 3-orbifolds, though. $\endgroup$
    – Marco Golla
    Commented Mar 4, 2023 at 10:00

2 Answers 2

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Finite subgroups of $\operatorname{SO}(4)$ and $\operatorname{O}(4)$ are enumerated in Chapter 4 of the book:

Conway, John H.; Smith, Derek A., On quaternions and octonions: their geometry, arithmetic, and symmetry, Natick, MA: A K Peters (ISBN 1-56881-134-9/hbk). xii, 159 p. (2003). ZBL1098.17001.

There may be results for some $n>4$, but since every finite group has faithful orthogonal representations, this gets hopeless at some point.

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There's a table here of all groups of order up to 500 with a 4-dimensional faithful orthogonal irreducible representation. This is almost the same thing as a subgroup of $O(4)$, but you'd also need to think about the reducible case.

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    $\begingroup$ This misses the most important ones (e.g., the perfect one of order $2.60^2=7200$). $\endgroup$
    – YCor
    Commented Mar 3, 2023 at 5:57

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