= Permutation wreath product
{title2=$H\wr_X K=H^X\rtimes K$}
If a <group> $K$ acts on a finite set $X$, it acts on the <direct product of groups> $H^X$ by permuting its factors. The resulting <semidirect product> is the permutation wreath product. For $K=S_m$ acting on $m$ two-point blocks and $H=C_2$, it describes independent within-block swaps combined with a permutation of blocks, and has order $2^m m!$. The action on $X$ is part of the definition; it need not be the regular action used in a regular <wreath product>.
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