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Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.

Is there any characterization of $\Gamma$ such that $\Gamma$ is conjugate in $\mathrm{O}(4)$ to a subgroup of $\mathrm{U}(2)$?

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    $\begingroup$ I think it helps if you give us the classification of $\Gamma $. $\endgroup$
    – user43326
    Commented Nov 5, 2021 at 10:06
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    $\begingroup$ @user43326, here it is en.wikipedia.org/wiki/Spherical_3-manifold $\endgroup$
    – Nick L
    Commented Nov 5, 2021 at 10:39
  • $\begingroup$ @NickL Thank you very much, it helps already, but they only say which group "can" act, but they don't say how these groups "do" act, or, in other words, how they are included in SO(3), or did I miss something? $\endgroup$
    – user43326
    Commented Nov 5, 2021 at 15:59
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    $\begingroup$ This is in Thurston's textbook, in the spherical 3-manifolds chapter. The precise embedding in $O_4$ is given. $\endgroup$
    – Ryan Budney
    Commented Nov 5, 2021 at 16:14
  • $\begingroup$ When I think about how groups act on spaces I usually think of an object which has this group as group of symmetry. Therefore your question reminded me of this answer in which Günter Rote describes a 4-polytope whose orientation-preserving symmetry group is $I\times C_7$ (one of the central extension of the icosahedral group mentioned on the Wikiepdia page). Since each of your groups $\Gamma$ is (abstractly) such a central extensions of a polyhedral group, an analogous construction should give you an object with symmetry group $\Gamma$. $\endgroup$
    – M. Winter
    Commented Nov 15, 2021 at 9:52

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