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I am looking for non-isomorphic finite groups that

  1. have the same order

  2. have faithful real irreducible representations

  3. the smallest dimension of faithful real irreducible representations are equal

Are there any examples, besides the cyclic and dihedral groups?

Are there any examples?


An equivalent question would be:

Are finite subgroups of $\operatorname{GL}(n,\mathbb{R})$ which are not subgroups of $\operatorname{GL}(m,\mathbb{R})$ for any $m<n$ uniquely determined by their order?

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The smallest example is the symmetric group $S_3$ and the cyclic group $\mathbb{Z}/6\mathbb{Z}$.

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  • $\begingroup$ Sorry, I accidentally deleted this during the most recent edit, but I was specifically looking for examples other than cyclic and dihedral groups. $\endgroup$ – WQ Arielle Cosette Mar 15 '18 at 16:58
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    $\begingroup$ Well, you can take a direct product of both with $\mathbb{Z}/2\mathbb{Z}$. These will be neither cyclic nor dihedral. $\endgroup$ – Alex B. Mar 15 '18 at 17:42

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