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Zbl 0792.20021
Glasby, S.P.; Howlett, R.B.
Extraspecial towers and Weil representations. (English)
[J] J. Algebra 151, No.1, 236-260 (1992). ISSN 0021-8693

An extraspecial tower is a finite group $G$ with a series of normal subgroups $G = N\sb 0 > N\sb 1 > N\sb 2 > \dots$ such that, for each $i$, $N\sb i/N\sb{i+1}$ is extraspecial and $N\sb{i+1}$ is contained in the derived group of $N\sb i$. Such towers provide examples of soluble groups of arbitrarily large derived length yet transparent structure. The authors construct them inductively ``from the top down'': given $G\sb i = G/N\sb i$, they choose a suitable $E\sb i = N\sb i/N\sb{i+1}$ and then construct $G\sb{i+1} = G/N\sb{i+1}$ as a split extension $G\sb{i + 1} = G\sb i \ltimes E\sb i$. The Weil representations are involved in the construction, whose details are quite complicated. An interesting example arising out of the paper is an iterated split extension $$Sp\sb 2(\bbfF\sb 3) \ltimes E \ltimes Q\sp 3 \ltimes Q\sp{81} \ltimes E\sp{2\sp{80}} \ltimes \dots,$$\vskip2.7mm where $Q$, $E$ denote respectively the quaternion group of order 8 and the nonabelian group of order 27 and exponent 3 and the powers are central (rather than direct) products.
[G.E.Wall (Sydney)]

MSC 2000:: *20D30 Series and lattices of subgroups of finite groups
20D15 Nilpotent finite groups
20E22 Extensions and other compositions of groups
20G40 Linear algebraic groups over finite fields
20G05 Representation theory of linear algebraic groups

Keywords: extraspecial $p$-groups; irreducible representations; symplectic groups; unitary groups; extraspecial tower; finite group; soluble groups; derived length; split extension; Weil representations

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