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On two groups of the form \(2^{8}{:}A_{9}\)

Abstract

This paper is dealing with two split extensions of the form \(2^{8}{:}A_{9}.\) We refer to these two groups by \(\overline{G}_{1}\) and \(\overline{G}_{2}.\) For \(\overline{G}_{1},\) the 8-dimensional GF(2)-module is in fact the deleted permutation module for \(A_{9}.\) We firstly determine the conjugacy classes of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of \(\overline{G}_{1}\) are \(A_{9},\,A_{8},\, S_{7},\,(A_{6} \times 3){:}2 \) and \((A_{5} \times A_{4}){:}2,\) while the inertia factor groups of \(\overline{G}_{2}\) are \(A_{9},\, PSL(2,8){:}3\) and \(2^{3}{:}GL(3,2).\) We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of \(\overline{G}_{1}\) and \(\overline{G}_{2}.\) The Fischer matrices of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of \(\overline{G}_{1}\) and \(\overline{G}_{2}\) are \(84 \times 84\) and \(40 \times 40\) complex valued matrices respectively.

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Acknowledgements

The authors would like to thank the referees for the corrections and valuable suggestions and comments, which improved the manuscript. The first author would like to thank his supervisor (second author) for his advice and support. Also the North-West University (Mafikeng) and the National Research Foundation (NRF) of South Africa are acknowledged for their financial supports.

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Correspondence to Ayoub B. M. Basheer.

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A. B. M. Basheer is currently a postdoctoral fellow at the North-West University, Mafikeng campus. J. Moori: support of the North-West University and National Research Foundation (NRF) of South Africa are acknowledged.

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Basheer, A.B.M., Moori, J. On two groups of the form \(2^{8}{:}A_{9}\) . Afr. Mat. 28, 1011–1032 (2017). https://doi.org/10.1007/s13370-017-0500-1

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Keywords

  • Group extensions
  • Alternating group
  • Inertia groups
  • Fischer matrices
  • Character table

Mathematics Subject Classification

  • 20C15
  • 20C40