Clifford-Fischer theory applied to a group of the form $2_{-}^{1+6}{:}((3^{1+2}{:}8){:}2)$

Document Type: Research Paper

Authors

1 School of Mathematical Sciences, ‎North-West University ‎(Mafikeng)‎, ‎P Bag X2046‎, ‎Mmabatho 2735‎, ‎South Africa.

2 School of Mathematical ‎Sciences, ‎North-West University ‎(Mafi-keng)‎, ‎P Bag X2046‎, ‎Mmabatho 2735‎, ‎South Africa.

Abstract

‎In our paper [A‎. ‎B‎. ‎M‎. ‎Basheer and J‎. ‎Moori‎, ‎On a group of the form $2^{10}{:}(U_{5}(2){:}2)$] we calculated the inertia factors‎, ‎Fischer matrices and the ordinary character table of the split‎ ‎extension $2^{10}{:}(U_{5}(2){:}2)$ by means of Clifford-Fischer‎ ‎Theory‎. ‎The second inertia factor group of $2^{10}{:}(U_{5}(2){:}2)$‎ ‎is a group of the form $2_{-}^{1+6}{:}((3^{1+2}{:}8){:}2).$ The‎ ‎purpose of this paper is the determination of the conjugacy classes‎ ‎of $\overline{G}$ using the coset analysis method‎, ‎the determination‎ ‎of the inertia factors‎, ‎the computations of the Fischer matrices and‎ ‎the ordinary character table of the split extension $\overline{G}=‎‎2_{-}^{1+6}{:}((3^{1+2}{:}8){:}2)$ by means of Clifford-Fischer‎ ‎Theory‎. ‎Through various theoretical and computational aspects we‎ ‎were able to determine the structures of the inertia factor groups‎. ‎These are the groups $H_{1} = H_{2} = (3^{1+2}{:}8){:}2,\$ $H_{3} =‎‎QD_{16}$ and $H_{4} = D_{12}.$ The Fischer matrices‎ ‎$\mathcal{F}_{i}$ of $\overline{G},$ which are complex valued‎ ‎matrices‎, ‎are all listed in this paper and their sizes range between‎ ‎2 and 5‎. ‎The full character table of $\overline{G},$ which is $41‎ ‎\times 41$ complex valued matrix‎, ‎is available in the PhD thesis of‎ ‎the first author‎, ‎which could be accessed online‎.

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References

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History

• Receive Date: 25 September 2014
• Revise Date: 21 February 2017
• Accept Date: 08 October 2015