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On the extension of \(G_2(3^{2n+1})\) by the exceptional graph automorphism. (English) Zbl 1138.20011

Summary: The main aim of this paper is to compute the character table of \(G_2(3^{2n+1})\rtimes\langle\sigma\rangle\), where \(\sigma\) is the graph automorphism of \(G_2(3^{2n+1})\) such that the fixed-point subgroup \(G_2(3^{2n+1})^\sigma\) is the Ree group of type \(G_2\). As a consequence we explicitly construct a perfect isometry between the principal \(p\)-blocks of \(G_2(3^{2n+1})^\sigma\) and \(G_2(3^{2n+1})\rtimes\langle\sigma\rangle\) for prime numbers dividing \(q^2-q+1\).

MSC:

20C33 Representations of finite groups of Lie type
20C20 Modular representations and characters
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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References:

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