ON A GROUP EXTENSION INVOLVING THE SPORADIC JANKO GROUP \(J_{2}\)

Ayoub B. M. Basheer     (School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P. Bag X1106, Sovenga 0727, South Africa; Mathematics Program, Faculty of Education, Sohar, Oman;, South Africa)

Abstract


According to the electronic Atlas [23], the group \(J_{2}\) has an absolutely irreducible module of dimension~6 over~\(\mathbb{F}_{4}.\) Therefore, a split extension group having the form \(4^{6}{:}J_{2}:= \overline{G}\) exists. In this paper, we consider this group. Our purpose is to determine its conjugacy classes and character table using the methods of the coset analysis together with Clifford--Fischer theory. We determine the inertia factors of \(\overline{G}\) by analyzing the maximal subgroups of \(J_{2}\) and the maximal of the maximal subgroups of \(J_{2}\) together with other various information. It turns out that the character table of  \(\overline{G}\) is a \(53 \times 53\) real-valued matrix, while Fischer matrices of the extension are all integer-valued matrices with sizes ranging from 1 to 8.


Keywords


Group extensions, Janko sporadic simple group, Inertia groups, Fischer matrices, Character table

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References


  1. Ali F., Moori J. The Fischer–Clifford matrices and character table of a maximal subgroup of \(Fi_{24}\). Algebra Colloq., 2010. Vol. 17, No. 3. P. 389–414. DOI: 10.1142/S1005386710000386
  2. Basheer A. B. M. Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups. PhD Thesis. Pietermaitzburg: University of KwaZulu-Natal, 2012.
  3. Basheer A.B.M. On a group involving the automorphism of the Janko group \(J_{2}\). J. Indones. Math. Soc., 2023. Vol. 29, No. 2. P. 197–216. DOI: 10.22342/jims.29.2.1371.197-216
  4. Basheer A.B.M. On a group extension involving the Suzuki group \(Sz(8)\). Afr. Mat., 2023. Vol. 34, No. 4. Art. no. 96. DOI: 10.1007/s13370-023-01130-z
  5. Basheer A.B.M., Moori J. Fischer matrices of Dempwolff group \(2^{5}{^{\cdot}}GL(5,2)\). Int. J. Group Theory, 2012. Vol. 1, No. 4. P. 43–63. DOI: 10.22108/IJGT.2012.1590
  6. Basheer A.B.M., Moori J. On the non-split extension group \(2^{6}{^{\cdot}}Sp(6,2)\). Bull. Iranian Math. Soc., 2013. Vol. 39, No. 6. P. 1189–1212.
  7. Basheer A.B.M., Moori J. On the non-split extension \(2^{2n}{^{\cdot}}Sp(2n,2)\). Bull. Iranian Math. Soc., 2015. Vol. 41, No. 2. P. 499–518.
  8. Basheer A.B.M., Moori J. On a maximal subgroup of the Thompson simple group. Math. Commun., 2015. Vol. 20, No. 2. P. 201–218. https://hrcak.srce.hr/149786
  9. Basheer A.B.M., Moori J. A survey on Clifford-Fischer theory. In: Groups St Andrews 2013, C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal (eds.). London Math. Soc. Lecture Note Ser., vol. 422. Cambridge University Press, 2015. P. 160–172. DOI: 10.1017/CBO9781316227343.009
  10. Basheer A.B.M., Moori J. On a group of the form \(3^{7}{:}Sp(6,2)\). Int. J. Group Theory, 2016. Vol. 5, No. 2. P. 41–59. DOI: 10.22108/IJGT.2016.8047
  11. Basheer A.B.M., Moori J. On two groups of the form \(2^{8}{:}A_{9}\). Afr. Mat., 2017. Vol. 28. P. 1011–1032. DOI: 10.1007/s13370-017-0500-1
  12. Basheer A.B.M., Moori J. On a group of the form \(2^{10}{:}(U_{5}(2){:}2)\). Ital. J. Pure Appl. Math., 2017. Vol. 37. P. 645–658. URL: https://ijpam.uniud.it/online\_issue/201737/57-BasheerMoori.pdf
  13. Basheer A.B.M., Moori J. Clifford-Fischer theory applied to a group of the form \(2^{1+6}_{-}{:}((3^{1+2}{:}8){:}2)\). Bull. Iranian Math. Soc., 2017. Vol. 43, No. 1. P. 41–52.
  14. Basheer A.B.M., Moori J. On a maximal subgroup of the affine general linear group \(GL(6,2)\). Adv. Group Theory Appl., 2021. Vol. 11. P. 1–30. DOI: 10.32037/agta-2021-001
  15. Bosma W., Cannon J.J. Handbook of Magma Functions, 1st ed. Sydney: University of Sydney, 1993. 690 p.
  16. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A. ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press, 1985. 250 p.
  17. Fray R.L., Monaledi R.L., Prins A.L. Fischer–Clifford matrices of \(2^{8}{:}(U_{4}(2){:}2)\) as a subgroup of \(O^{+}_{10}(2)\). Afr. Mat., 2016. Vol. 27. P. 1295–1310. DOI: 10.1007/s13370-016-0410-7
  18. GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. Version 4.4.10, 2007. URL: http://www.gap-system.org
  19. Maxima, A Computer Algebra System. Version 5.18.1, 2009. URL: http://maxima.sourceforge.net
  20. Moori J. On the Groups \(G^{+}\) and \(\overline{G}\) of the Form \(2^{10}{:}M_{22}\) and \(2^{10}{:}\overline{M}_{22}\). PhD Thesis. Birmingham: University of Birmingham, 1975.
  21. Moori J. On certain groups associated with the smallest Fischer group. J. London Math. Soc., 1981. Vol. 2. P. 61–67.
  22. Wilson R.A. The Finite Simple Groups. London: Springer-Verlag, 2009. XV, 298 p. DOI: 10.1007/978-1-84800-988-2
  23. Wilson R.A., et al. Atlas of Finite Group Representations. Version 3. URL: http://brauer.maths.qmul.ac.uk/Atlas/v3/




DOI: http://dx.doi.org/10.15826/umj.2024.1.003

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