AUT Journal of Mathematics and Computing

AUT Journal of Mathematics and Computing

On an imprimitive maximal subgroup of $SU_7(2)$

Document Type : Original Article

Authors
David Mwanzia Musyoka 1
Abraham Love Prins 2
Lydia Nyambura Njuguna 3
Lucy Chikamai 4
1 Department of Mathematics, Statistics and Actuarial Sciences Karatina University P. O. Box 1957-10101, Kenya
2 Department of Computational Sciences, University of Fort Hare, Private Bag X1314, King William’s Town Road, Alice, 5700, South Africa
3 Department of Mathematics and Actuarial Science, Kenyatta University, PO Box 43844 - 00100, Nairobi, Kenya
4 Department of Mathematics and Actuarial Science, Kibabii University, PO Box 1699 - 50200, Bungoma, Kenya
10.22060/ajmc.2025.23932.1340
Abstract
The special unitary group $SU_{n}(q)$ has a maximal imprimitive subgroup with the structure $(q+1)^{n-1}{:}S_{n}$. The exceptions when the subgroup is not maximal are $SU_{3}(5)$, $SU_{4}(3)$ and $SU_{6}(2)$. In this paper, the ordinary character table of the maximal imprimitive subgroup $\overline{G}=3^{6}{:}S_{7}$ of $SU_{7}(2)=U_{7}(2)$ is computed by the Fischer-Clifford matrices technique. A combinatorial approach is adopted in the computation of the Fischer-Clifford matrices of $\overline{G}$.
Keywords
Character table
Fischer-Clifford matrix
Inertia factor
Coset analysis
Subjects
[1]      M. S. O. Almestady, Fischer Matrices for Generalised Symmetric Groups–A Combinatorial Approach, PhD thesis, University of Wales, Aberystwyth, 1998.
[2]      A. B. M. Basheer and J. Moori, A survey on Clifford-Fischer theory, in Groups St Andrews 2013, vol. 422 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2015, pp. 160–172.
[3]      J. N. Bray, D. F. Holt, and C. M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, vol. 407 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2013. With a foreword by Martin Liebeck.
[4]      C. Chileshe, Irreducible characters of Sylow p-Subgroups associated with some classical linear groups, PhD thesis, North-West University, South Africa, 2016.
[5]      C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962.
[6]      B. Fischer, Clifford-matrices, in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), vol. 95 of Progr. Math., Birkha¨user, Basel, 1991, pp. 1–16.
[7]      GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.2, 2022. Version 4.12.2.
[8]      D. Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968.
[9]      P. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups, vol. 129 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1990.
[10]  R. J. List, On the characters of 2n−ϵ · Sn, Arch. Math. (Basel), 51 (1988), pp. 118–124.
[11]  R. J. List and I. M. I. Mahmoud, Fischer matrices for wreath products GwSn, Arch. Math. (Basel), 50 (1988), pp. 394–401.
[12]  J. Moori and T. Seretlo, On the Fischer-Clifford matrices of a maximal subgroup of the Lyons group Ly, Bull. Iranian Math. Soc., 39 (2013), pp. 1037–1052.
[13]  J. Moori and T. Seretlo, On 2 nonsplit extension groups associated with HS and HS:2, Turkish J. Math., 38 (2014), pp. 60–78.
[14]  Z. E. Mpono, Fischer-Clifford Theory and Character Tables of Group Extensions, PhD thesis, University of Natal, Pietermaritzburg, 1998.
[15]  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.
[16], The character table of an involution centralizer in the Dempwolff group 25·GL5(2), Quaest. Math., 39 (2016), pp. 561–576.
[17], A maximal subgroup 24+6:(A5 × 3) of G2(4) treated as a non-split extension G = 26·(24:(A5 × 3)), Adv. Group Theory Appl., 10 (2020), pp. 43–66.
[18]  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), Afr. Mat., 31 (2020), pp. 1311–1336.
[19]  R. P. Stanley, Enumerative combinatorics. Volume 1, vol. 49 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second ed., 2012.
[20]  N. S. Whitley, Fischer matrices and character tables of group extensions, Master’s thesis, 1994.
[21]  R. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray, and R. Abbott, Atlas of finite group representation. http://brauer.maths.qmul.ac.uk/Atlas/v3/. http://brauer.maths.qmul.ac.uk/Atlas/v3/.
[22]  K. Zimba, Fischer-Clifford Matrices of the Generalized Symmetric Group and some Associated Groups, PhD thesis, University of KwaZulu-Natal, Pietermaritzburg, 2005.
AUT Journal of Mathematics and Computing
Volume 7, Issue 4
Autumn 2026
Pages 423-440