Timeline for Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below

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Dec 14, 2018 at 17:45	comment	added	Guntram		You need to exclude the edge case that $N$ is abelian (e.g., $SL_n$ works for $n \geq 3$). I think more generally $H$ can be taken to be a Levi factor of a parabolic subgroup $P=L \ltimes U$ with the same caveat (your example is the case $P=B = T\ltimes N$).
Dec 13, 2018 at 19:06	history	answered	Victor Petrov	CC BY-SA 4.0