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Abstract
The Janko sporadic simple group J2 has an automorphism group 2. Using the electronic Atlas of Wilson [22], the group J2:2 has an absolutely irreducible module of dimension 12 over F2. It follows that a split extension group of the form 2^12:(J2:2) := G exists. In this article we study this group, where we compute its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. The inertia factor groups of G will be determined by analysing the maximal subgroups of J2:2 and maximal of the maximal subgroups of J2:2 together with various other information. It turns out that the character table of G is a 64×64 real valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 6.
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References
- Ali F. and Moori J., The Fischer-Clifford matrices and character table of a maximal subgroup of F i24, Algebra Colloquium, 17 No. 3 (2010), 389-414.
- Basheer A.B.M., Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups, PhD Thesis, University of KwaZulu-Natal, Pieter-maitzburg, 2012, 2012.
- Basheer A.B.M. and Moori J., Fischer matrices of Dempwolff group 25·GL(5, 2), International Journal of Group Theory, 1 No. 4 (2012), 43-63.
- Basheer A.B.M. and Moori J., On the non-split extension group 26·Sp(6, 2), Bulletin of the Iranian Mathematical Society, 39 No. 6 (2013), 1189-1212.
- Basheer A.B.M. and Moori J., On the non-split extension 22n·Sp(2n, 2), Bulletin of the Iranian Mathematical Society, 41 No. 2 (2015), 499-518.
- Basheer A.B.M. and Moori J., On a maximal subgroup of the Thompson simple group, Mathematical Communications, 20 No. 2 (2015), 201-218.
- Basheer A.B.M. and Moori J., A survey on Clifford-Fischer theory, London Mathematical Society Lecture Notes, 422, published by Cambridge University Press (2015), 160-172.
- Basheer A.B.M. and Moori J., On a group of the form 37:Sp(6, 2), International Journal of Group Theory, 5 No. 2 (2016), 41-59.
- Basheer A.B.M. and Moori J., On two groups of the form 28:A9, Afrika Matematika, 28 (2017), 1011-1032.
- Basheer A.B.M. and Moori J., On a group of the form 210:(U5(2):2), Italian Journal of Pure and Applied Mathematics, 37 (2017), 645-658.
- Basheer A.B.M. and Moori J., Clifford-Fischer theory applied to a group of the form 2 1+6− :((31+2:8):2), Bulletin of the Iranian Mathematical Society, 43, No. 1 (2017), 41-52.
- Basheer A.B.M. and Moori J., On a maximal subgroup of the affine general linear group AGL(6, 2), Advances in Group Theory and Applications, 11 (2021), 1-30.
- Bosma W. and Cannon J.J., Handbook of Magma Functions, Department of Mathematics, University of Sydeny, 1994.
- Chileshe C., Moori J., and Seretlo T.T., On a aaximal parabolic subgroup of O+8(2), Bulletin of the Iranian Mathematical Society, 44 (2018), 159181.
- Conway J.H., Curtis R.T., Norton S.P., Parker R.A., and Wilson R.A., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
- Fray R.L., Monaledi R.L., and Prins A.L., Fischer-Clifford matrices of 28:(U4(2) : 2) as a subgroup of O+10(2), Afrika Matematika, 27 (2016), 12951310.
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10; 2007. (http://www.gap-system.org)
- Maxima, A Computer Algebra System. Version 5.18.1; 2009. (http://maxima.sourceforge.net)
- Moori J., On the Groups G+ and G of the form 210:M22 and 210:M22, PhD Thesis, University of Birmingham, 1975.
- Moori J., On certain groups associated with the smallest Fischer group, J. London Math. Soc., 2 (1981), 61-67.
- Wilson R.A., The Finite Simple Groups, Springer-Verlag London Limited, 2009.
- Wilson R.A. and et al., Atlas of finite group representations, (http://brauer.maths.qmul.ac.uk/Atlas/v3/)
References
Ali F. and Moori J., The Fischer-Clifford matrices and character table of a maximal subgroup of F i24, Algebra Colloquium, 17 No. 3 (2010), 389-414.
Basheer A.B.M., Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups, PhD Thesis, University of KwaZulu-Natal, Pieter-maitzburg, 2012, 2012.
Basheer A.B.M. and Moori J., Fischer matrices of Dempwolff group 25·GL(5, 2), International Journal of Group Theory, 1 No. 4 (2012), 43-63.
Basheer A.B.M. and Moori J., On the non-split extension group 26·Sp(6, 2), Bulletin of the Iranian Mathematical Society, 39 No. 6 (2013), 1189-1212.
Basheer A.B.M. and Moori J., On the non-split extension 22n·Sp(2n, 2), Bulletin of the Iranian Mathematical Society, 41 No. 2 (2015), 499-518.
Basheer A.B.M. and Moori J., On a maximal subgroup of the Thompson simple group, Mathematical Communications, 20 No. 2 (2015), 201-218.
Basheer A.B.M. and Moori J., A survey on Clifford-Fischer theory, London Mathematical Society Lecture Notes, 422, published by Cambridge University Press (2015), 160-172.
Basheer A.B.M. and Moori J., On a group of the form 37:Sp(6, 2), International Journal of Group Theory, 5 No. 2 (2016), 41-59.
Basheer A.B.M. and Moori J., On two groups of the form 28:A9, Afrika Matematika, 28 (2017), 1011-1032.
Basheer A.B.M. and Moori J., On a group of the form 210:(U5(2):2), Italian Journal of Pure and Applied Mathematics, 37 (2017), 645-658.
Basheer A.B.M. and Moori J., Clifford-Fischer theory applied to a group of the form 2 1+6− :((31+2:8):2), Bulletin of the Iranian Mathematical Society, 43, No. 1 (2017), 41-52.
Basheer A.B.M. and Moori J., On a maximal subgroup of the affine general linear group AGL(6, 2), Advances in Group Theory and Applications, 11 (2021), 1-30.
Bosma W. and Cannon J.J., Handbook of Magma Functions, Department of Mathematics, University of Sydeny, 1994.
Chileshe C., Moori J., and Seretlo T.T., On a aaximal parabolic subgroup of O+8(2), Bulletin of the Iranian Mathematical Society, 44 (2018), 159181.
Conway J.H., Curtis R.T., Norton S.P., Parker R.A., and Wilson R.A., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
Fray R.L., Monaledi R.L., and Prins A.L., Fischer-Clifford matrices of 28:(U4(2) : 2) as a subgroup of O+10(2), Afrika Matematika, 27 (2016), 12951310.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10; 2007. (http://www.gap-system.org)
Maxima, A Computer Algebra System. Version 5.18.1; 2009. (http://maxima.sourceforge.net)
Moori J., On the Groups G+ and G of the form 210:M22 and 210:M22, PhD Thesis, University of Birmingham, 1975.
Moori J., On certain groups associated with the smallest Fischer group, J. London Math. Soc., 2 (1981), 61-67.
Wilson R.A., The Finite Simple Groups, Springer-Verlag London Limited, 2009.
Wilson R.A. and et al., Atlas of finite group representations, (http://brauer.maths.qmul.ac.uk/Atlas/v3/)