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On the wikipedia page of tetrahedron, there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces). There is also a list on the page of 5-cell but doesn't seem complete.

This seems elementary, but I can not find a source to cite. Also, is there already a general classification for symmetry groups of unmarked (possibly irreglar) d-simplices (or an algorithm for enumerating it)?

I believe this is equivalent to asking about the symmetry of edge-colored complete graphs. I find a lot of literature on those with transitive groups, but not general cases.

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  • $\begingroup$ As a help for others to find an answer: this seems an interesting and perfectly well-defined question; to paraphrase: the OP is asking for citable sources on what are all the possible isomorphism types of of the (necessarily finite) subgroups which arise as the set-wise stabiliser, inside the full Euclidean group $E(n)$, of a specified geometric realization of a simplicial complex in $\mathbb{R}^n$, the realization being arbitrary (except, perhaps, being piecewise-linear, but even that needn't be said, I think). There are indeed [...] $\endgroup$
    – Peter Heinig
    Commented Oct 7, 2017 at 9:27
  • $\begingroup$ [...] many such isomorphism groups, starting from the trivial group (which arises for most 'irregular' geometric realizations) all the way to the symmetric group $\mathrm{Sym}(n)$. The OP also points out that any such geometric realization defines an inclusion of groups of the relevant symmetryi group into the autormorphism group of a $k$-colored graph in the sense of e.g. [Mariusz Grech, Andrzej Kisielewicz: All totally symmetric colored graphs. arXiv:1201.4464v1]. The OP seems right in that it is not easy to read-off an answer to the question from the literature, in particular [...] $\endgroup$
    – Peter Heinig
    Commented Oct 7, 2017 at 9:35
  • $\begingroup$ [...] op. cit. has a general title, yet is not sufficient by itself to answer this question. The most difficult part of the question (to me) seems to rigorously show that the question even can be reduced to graph theory; it is not clear that all the relevant graph-theoretic symmetries can be realized by a Euclidean symmetry. It seems that if someone likes to get into this question, then carefully working out the connection between coloring-preserving automorphisms of complete graphs and simplex-realization-preserving Euclidean symmetries is the most important step towards an answer. $\endgroup$
    – Peter Heinig
    Commented Oct 7, 2017 at 9:39
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    $\begingroup$ FYI, I consulted with the wikipedia editor. He used a software to calculate the subgroups of the tetrahedral group. He also mentioned Conway's book "the symmetries of things". There is a poset for the octahedral group, from which one can extract a poset for the tetrahedral group (if he manages to understand the notation system). $\endgroup$
    – Hao Chen
    Commented Oct 13, 2017 at 18:41
  • $\begingroup$ But all these are only for dimension 3 and are not direct reference for what I'm asking. $\endgroup$
    – Hao Chen
    Commented Oct 13, 2017 at 18:42

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