= Order-sixteen Gassmann pair
{title2=$C_4\times C_4,\ Q_8\times C_2\leq S_{16}$}
Embed $C_4\times C_4$ and $Q_8\times C_2$ in $S_{16}$ by their <regular representations>. Both have one element of order one, three of order two, and twelve of order four. An order-$r$ element in a regular action has cycle type $r^{16/r}$. Their intersections with every <conjugacy class> of $S_{16}$ consequently have the same size, so they are <Gassmann equivalent>. The first group is abelian and the second is not, so their underlying groups are not isomorphic. Finite-cover constructions from the <Sunada theorem> therefore give <isospectral manifolds> with different <fundamental groups>.
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