I am looking for a family of finite solvable groups (with infinite members) with given presentations. I am aware that for example dihedral groups, or dicyclic groups have well-known presentations. Also, certain 2-groups have been constructed using generators and relations. However, I need a family of whose members are solvable but not supersolvable. Any reference to such structures will be appreciated.
2026-03-26 20:44:16.1774557856
presentation of a family of solvable groups
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Let $X$ be a finite solvable group, and let $V$ be an irreducible $F[X]$-module over some finite field $F$. Then in most cases the affine group $V \rtimes X$ is a finite solvable group which is not supersolvable. (Since $V$ is minimal normal in $V \rtimes X$, and usually not cyclic).
Given generators and relations for $X$ and the action of generators of $X$ on $V$, you can construct a presentation of $V \rtimes X$.