= Nonisomorphic Gassmann equivalent regular subgroups
{title2=$(C_p)^3,\ UT_3(\mathbb F_p)\leq S_{p^3}$}
For an odd prime $p$, embed the elementary abelian group $(C_p)^3$ and the <Heisenberg group over a prime field> regularly in $S_{p^3}$. Every nonidentity element has order $p$ and hence regular permutation cycle type $p^{p^2}$. The two <subgroups> have equal intersection counts with every symmetric-group <conjugacy class>, so they are <Gassmann equivalent>, but they are not isomorphic because one is abelian and the other is not. Using a compact <simply connected> isometric cover with ambient deck group $S_{p^3}$ gives quotients with these nonisomorphic <fundamental groups>; <Sunada theorem> makes them isospectral while their topology differs.
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