On a maximal subgroup of the orthogonal group O⁺₈ (3)
DOI:
https://doi.org/10.22199/issn.0717-6279-4778Keywords:
Coset analysis, Fischer-Clifford matrices, split extension, inertia factor, character table, fusion map, restriction of charactersAbstract
The orthogonal simple group 0 (3) has three conjugacy classes of maximal subgroups of the form 36:L4(3). These groups are all isomorphic to each other and each group has order 4421589120 with index 1120 in 0 (3). In this paper, we will compute the ordinary carácter table of one of these classes of maximal subgroups using the technique of Fischer-Clifford matrices. This technique is very efficient to compute the ordinary character table of an extension group Ḡ = N.G and especially where the normal subgroup N of Ḡ is an elementary abelian p-group. The said technique reduces the computation of the ordinary character table of Ḡ to find a handful of so-called Fischer-Clifford matrices of Ḡ and the ordinary or projective character tables of the inertia factor groups of the action of Ḡ on N.
References
F. Ali, Fischer-Clifford theory for split and non-split group extensions, PhD Thesis, University of Natal, 2001.
A. B. M. Basheer, J. Moori, and C. M. Roney-Dougal, “A survey on Clifford-Fischer theory,” in Groups St Andrews 2013, C. M. Campbell, M. R. Quick, and E. F. Robertson , Eds. Cambridge: Cambridge University Press, 2015, pp. 160–172.
A. B. Basheer and T. T. Seretlo, “On a group of the form 214:Sp(6, 2)”, Quaestiones mathematicae, vol. 39, no. 1, pp. 45–57, 2015. https://doi.org/10.2989/16073606.2015.1023865
A. Basheer and J. Moori, “On a group of the form 37:Sp(6,2)”, International journal of group theory, vol. 5, no. 2, pp. 41-59, 2016. https://doi.org/10.22108/IJGT.2016.8047
C. Chileshe, Irreducible characters of Sylow p-Subgroups associated with some classical linear groups, PhD. Thesis, North-West University, 2016.
O. Bonten, Clifford-Matrizen, Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, 1988.
W. Bosma and J. J. Canon (Eds.), Handbook of magma functions, Sidney: University of Sydney, 1994.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups, Oxford : Oxford University Press, 1985.
B. Fischer, “Clifford-matrices”, in Representation Theory of Finite Groups and Finite-Dimensional Algebras, G. O. Michler and C. M. Ringel. Eds. Birkhäuser: Springer, pp. 1-16.
P. X. Gallagher, “Group characters and normal hall subgroups”, Nagoya mathematical journal, vol. 21, pp. 223–230, 1962. https://doi.org/10.1017/s0027763000023849
“GAP system for computational discrete algebra. version 4.11.0,” GAP System for Computational Discrete Algebra, Feb-2020. [Online]. Available: https://www.gap-system.org/Releases/4.11.0.html
I. M. Isaacs, Character theory of finite groups, San Diego: Academic Press, 1976.
R. J. List, “On the characters of 2n- ɛ Sn”, Archiv der mathematik, vol. 51, no. 2, pp. 118–124, 1988. https://doi.org/10.1007/bf01206468
R. J. List and I. M. I. Mahmoud, “Fischer matrices for wreath products GwSn”, Archiv der mathematik, vol. 50, no. 5, pp. 394–401, 1988. https://doi.org/10.1007/bf01196499
J. Moori and T. Seretlo, “On 2 nonsplit extension groups associated with HS and HS:2”, Turkish journal of mathematics, vol. 38, pp. 60–78, 2014. https://doi.org/10.3906/mat-1209-11
J. Moori and T. Seretlo, “On the Fischer-Clifford matrices of a maximal subgroup of the Lyons group Ly”, Bulletin of the Iranian mathematical Society, vol. 39, no. 5, pp. 1037-1052, 2013. [On line]. Available: https://bit.ly/3GIAqc6
J. Moori, On the Groups G+ and Ḡ of the forms 210:M22 and 210:M22, PhD. thesis, University of Birmingham, 1975.
Z. E. Mpono, Fischer-Clifford theory and character tables of group extensions, PhD. thesis, University of Natal, 1998.
A.L. Prins, “A maximal subgroup 24+6:(A5 × 3) of G2(4) treated as a non-split extension Ḡ = 26·(24:(A5 × 3))”, Advances in group theory and applications, vol. 10, pp. 43-66, 2020.https://doi.org/10.32037/agta-2020-009
A. L. Prins, R. L. Monaledi, and R. L. Fray, “On a maximal subgroup (29:(L3(4)):3 of the automorphism group U6(2):3 of U6(2)”, Afrika matematika, vol. 31, no. 7-8, pp. 1311–1336, 2020. https://doi.org/10.1007/s13370-020-00798-x
A. L. Prins, “The character table of an involution centralizer in the Dempwolff group 25·GL5(2)”, Quaestiones mathematicae, vol. 39, no. 4, pp. 561–576, 2015. https://doi.org/10.2989/16073606.2015.1113210
A. L. Prins, 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, 2011.
R. B. Salleh, On the construction of the character tables of extensión groups, PhD. thesis, University of Birmingham, 1982.
T. T. Seretlo, Fischer Clifford Matrices and Character Tables of Certain Groups Associated with Simple Groups 0+ 10 (2), HS and Ly, PhD. thesis, University of KwaZulu Natal, 2011.
R. A. Wilson, P. Walsh, J. Tripp, I. Suleiman, S. Rogers, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray and R. Abbot, ATLAS of finite group representations. [On line]. Available: http://brauer.maths.qmul.ac.uk/Atlas/v3/
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