The Fischer-Clifford matrices of the inertia group $2^7{:}O^{-}_{6}(2)$ of a maximal subgroup $2^7{:}Sp_6(2)$ in $Sp_8(2)$

Document Type: Research Paper

Authors

1 Stellenbosch University

2 University of the Western Cape

Abstract

The subgroups of symplectic groups which fix a non-zero vector of the underlying symplectic space are called affine subgroups., The split extension group $A(4)\cong 2^7{:}Sp_6(2)$ is the affine subgroup of the symplectic group $Sp_8(2)$ of index $255$‎. ‎In this paper‎, ‎we use the technique of the Fischer-Clifford matrices to construct the character table of the inertia group $2^7{:}O^{-}_{6}(2)$ of $A(4)$ of index $28$‎.

Keywords

Main Subjects


Faryad Ali (2001). Fischer-Clifford Theory and Character Tables of Group Extensions. PhD Thesis, University of Natal.
F. Ali and J. Moori (2004). Fischer-Clifford matrices and character table of the group $2^7:Sp_6(2)$. Int. J. Math. Game Theory and Algebra. 14, 101-121
W. Bosma and J. J. Canon (1994). Handbook of Magma Functions. Department of Mathematics, University of Sydney, November.
J. J. Cannon (1984). An introduction to the group theory language CAYLEY. Computational Group Theory (M. Atkinson, eds), Academic Press, San Diego. , 145-183
J. H. Conway, et al (1985). Atlas of Finite Groups. Oxford University Press, Oxford.
B. Fischer (1991). Clifford-matrices. Progr. Math., Michler G. O. and Ringel C.(eds), Birkhauser, Basel. 95, 1-16
D. Gorenstein (1968). Finite Groups. Harper and Row Publishers, New York.
I. M. Isaacs (1976). Character Theory of Finite Groups. Academic Press, San Diego.
G. Karpilovsky (1992). Group Representations: Introduction to Group Representations and Characters. Part B, North - Holland Mathematics Studies 175, Amsterdam. 1
J. Moori (1981). On certain groups associated with the smallest Fischer group. J. London Math. Soc. (2). 23, 61-67
J. Moori (1975). On the Groups $G^+$ and $overline{G}$ of the forms $2^{10}:M_{22}$ and $2^{10}:overline{M}_{22}$. PhD thesis, University of Birmingham.
J. Moori and Z. E. Mpono (2000). Fischer-Clifford matrices and the character table of a maximal subgroup of $overline{F}_{22}$. Int. J. Math. Game Theory Algebra. 10, 1-12
Z. Mpono (1998). Fischer-Clifford Theory and Character Tables of Group Extensions. PhD Thesis, University of Natal.
A. L. Prins (2011). Fischer-Clifford Matrices and Character Tables of Inertia Groups of Maximal Subgroups of Finite Simple Groups of Extension Type. PhD Thesis, University of the Western Cape.
M. Schonert, et al (1992). GAP - Groups, Algorithms and Programming. Lehrstul D Fur Matematik, RWTH-Aachen.
N. S. Whitley (1994). Fischer Matrices and Character Tables of Group Extensions. MSc Thesis, University of Natal.
K. Zimba (2005). Fischer-Clifford Matrices of the Generalized Symmetric Group and some Associated Groups. PhD Thesis, University of KwaZulu Natal.