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Sufficient conditions would establish the existence of a subgroup
stabilizing a point in the permutation representation. 
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One cannot do this just from the character table: e.g. $D_8$ and $Q_8$ have the same character table but different permutation characters; if $C_2<D_8$ is non-central, then $C[D_8/C_2]$ has character $(4,0,2,0,0)$ which is not a permutation character for $Q_8$. – Tim Dokchitser Nov 15 '11 at 13:02

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