Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is 16 the minimum?
2026-03-28 01:18:48.1774660728
Quaternion^2 in the symmetric group
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In general we have $\mu(G\times H)\le \mu(G)+\mu(H)$ for the minimal faithful permutation degree $\mu$. There are examples known with strict inequality, see here. However, for $p$-groups we have equality. That is, we have $\mu(Q_8\times Q_8)=\mu(Q_8)+\mu(Q_8)=8+8=16$.