Non-isomorphic finite groups with the same order and minimal dimension of real faithful irreducible representation

Question

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?

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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?

Answer 1

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