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  "body": "\n## Summary\n\nThe wreath product $A \\wr_\\Omega G$ builds a new group from a group $A$ and a group $G$ that acts on a set $\\Omega$.\nIt is the [[semidirect product|0H3WZXCZNF]] of the group of all functions $\\Omega \\to A$ with $G$, where $G$ moves the functions by moving the points of $\\Omega$.\nWreath products describe the symmetries of objects made of several copies of a smaller object.\nEach [[group extension|MF4N3VGGFV]] $E$ of a group $B$ by a group $A$ embeds in the unrestricted regular wreath product $A \\wr B$.\n\n## Explanations\n\n### L1 human\n\nThink of $m$ identical boxes, each holding a copy of some object with symmetry group $A$.\nA symmetry of the whole picture may turn each box by its own element of $A$, and then permute the boxes by an element of $G$.\nThe wreath product is the group of all such combined moves.\nFor example, take two coins on a table, each of which can be flipped, and allow the two coins to swap places.\nEach coin gives a group of order $2$, the swap gives another group of order $2$, and the group of all moves is $C_2 \\wr C_2$, of order $2^2 \\cdot 2 = 8$.\n",
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