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LSpice
LSpice
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Number of conjugacy classclasses of a semi-direct product of two finite groupgroups

DenoteLet $G$ and $H$ arebe two finite groups. Let $r(G)$ be the order of the set of conjugacy classclasses of G$G$. We know $$r(G\times H)=r(G)\times r(H).$$My My problem is that: if there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decomposedecomposed in the form $G\times H$,then if then do we have $$r(G\rtimes H)<r(G\times H)?$$

conjugacy class of semi-direct product of two finite group

Denote $G$ and $H$ are two finite groups. Let $r(G)$ be the order of the set of conjugacy class of G. We know $$r(G\times H)=r(G)\times r(H).$$My problem is that there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decompose the form $G\times H$,then if $$r(G\rtimes H)<r(G\times H)?$$

Number of conjugacy classes of a semi-direct product of two finite groups

Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decomposed in the form $G\times H$, then do we have $$r(G\rtimes H)<r(G\times H)?$$

added a top-level tag; see: https://meta.mathoverflow.net/questions/1457/why-are-mo-tags-formatted-as-they-are
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Martin Sleziak
Martin Sleziak
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gdre
gdre
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conjugacy class of semi-direct product of two finite group

Denote $G$ and $H$ are two finite groups. Let $r(G)$ be the order of the set of conjugacy class of G. We know $$r(G\times H)=r(G)\times r(H).$$My problem is that there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decompose the form $G\times H$,then if $$r(G\rtimes H)<r(G\times H)?$$