How does one compute the lowest dimensional faithful representation of a finite group?

This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape whose rotational/reflection symmetries form $G$. (Formally stated as finding the lowest dimensional faithful representation of $G$ into the orthogonal group $O(n)$.)

Now just listing out examples is hard, because even in just $\mathbb{R}^2$ outside of the symmetries of simple polygons we see things like $\mathbb{Z}_2 \times \mathbb{Z}_2 $ crop up, yet it appears that $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ doesn't have any obvious two dimensional representations. Grouppropswiki doesn't even have a single representation for it at all.

So I thought it was wise before I tackle the orthogonal group question I should know how to just, in general, find a low dimensional faithful representation. Some trivialities are that the dimension for a group $G$ will be less than or equal to the smallest $j$ such that $G \subset S_j$ since each symmetric group can be realized as the symmetries of a $j$-dimensional simplex space due to Burnside.

But this doesn't say much because even something like $\mathbb{D}_{\text{Graham's Number}}$ can be realized in $\mathbb{R}^2$ (and we still are dealing with isometries here, what about represenatations that transcend that!?)

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    $\begingroup$ Are you talking about complex representations, or representations over an arbitrary field? For complex representations, the smallest dimension of a faithful representation of a finite abelian group with minimal generating set of size $d$ is $d$. But $C_2 \times C_2 \times C_2$ has a faithful $2$-dimensional representation over the finite field of order $8$. In general, this is a difficult problem - even finding the smallest degree faithful permutation representation can be difficult. $\endgroup$
    – Derek Holt
    Mar 8, 2016 at 8:43
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    $\begingroup$ I'm focusing on real representations. Does that make matters worse? $\endgroup$ Mar 8, 2016 at 9:04
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    $\begingroup$ It's a difficult problem in general over any field. For an abelian group with invariant factor decomposition $C_{k_1} \times \cdots C_{k_r} \times C_2^s$ with each $k_i > 2$, I think the smallest dimension of a real faithful representation is $2r+s$. It is certainly $s$ for $C_2^s$. $\endgroup$
    – Derek Holt
    Mar 8, 2016 at 9:16
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    $\begingroup$ Is there a list of these smallest dimensions somewhere for small groups? @DerekHolt $\endgroup$ Mar 1, 2018 at 10:26
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    $\begingroup$ For future visitors: some research developments since this question was asked mathoverflow.net/questions/351938/… as well it is interesting to ask even whether there are small irreducible representations, as discussed here: mathoverflow.net/questions/400864/… Finally, note that $j$ in the question statement is called the minimal degree in group theory; computing is still open in many cases. $\endgroup$ Apr 13, 2022 at 4:19


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