A Maximal Subgroup 2^{4+6}:(A_5 x 3) of G_2(4) Treated as a Non-Split Extension \overline{G} = 2^{6·}(2^4:(A_5 x 3))

• Main Author: Abraham Love Prins
• Format: Article
• Language: English
• Published: Aracne 2020-12-01
• Series: Advances in Group Theory and Applications
• Subjects: projective character table; schur multiplier; fischer-clifford matrix
• Online Access: http://www.advgrouptheory.com/journal/Volumes/10/Prins.pdf
• ISSN: 2499-1287; 2499-1287

The maximal subgroup $2^{4+6}{:}(A_5\times3)$ of the Chevalley group $G_2(4)$ is isomorphic to a non-split extension group of the shape $\overline{G}=2^{6}{{}^\cdot}(2^4{:}(A_5\times3))$. In this paper, the ordinary character table of $2^{4+6}{:}(A_5\times3)$ will be re-calculated using the technique of Fischer-Clifford matrices and where $2^{4+6}{:}(A_5\times3)$ will be treated as the \hbox{non-split} extension $\overline{G}=2^{6}{{}^\cdot}(2^4{:}(A_5\times3))$. The author uses some relevant techniques to identify and compute the type of characters (ordinary or projective ) of the inertia factor groups $H_i$ of $\overline{G}$ on $\textnormal{Irr}(2^6)$, which are required in the construction of the character table of $\overline{G}$ via Fischer-Clifford theory. Also, this is a very good example to demonstrate how to apply Fischer-Clifford theory to a non-split extension group~$N{{}^\cdot}G$, where not every irreducible character of $N$ can be extended to its inertia group~$\overline{H}_i$ in $N{{}^\cdot}G$.