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On a maximal subgroup (29 :(L3 (4)):3 of the automorphism group U6 (2):3 of U6 (2) Abraham Love Prins1

· Ramotjaki Lucky Monaledi2 · Richard Llewellyn Fray3

Received: 16 October 2018 / Accepted: 5 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020

Abstract In this paper, the Fischer–Clifford matrices and associated character table of a maximal subgroup (29 :(L 3 (4)):3 of one of the automorphism groups U6 (2):3 of the unitary group U6 (2) are constructed. Keywords Coset analysis · Fischer–Clifford matrices · Permutation character · Character table · Inertia group · Set intersection Mathematics Subject Classification 20C15 · 20C40

1 Introduction The unitary group U6 (2) has three automorphism groups of the forms U6 (2):2, U6 (2):3 and U6 (2):S3 (see [1–3]). In this paper, the focus will be on one of the maximal subgroups G of U6 (2):3 with structure description (29 :L 3 (4)):3. Firstly, we will show with the help of MAGMA [4] that the group G = (29 :L 3 (4)):3 can be represented as a split extension G = 29 :(L 3 (4):3), where we regard 29 as the vector space V9 (2) and upon which L 3 (4):3 acts absolutely irreducibly as a matrix group G of dimension 9 over the Galois field G F(2). Since the action of G on its natural module V9 (2) is absolutely irreducible, the group G exists as subgroup of S L 10 (2).

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Abraham Love Prins [email protected] Ramotjaki Lucky Monaledi [email protected] Richard Llewellyn Fray [email protected]

1

Department of Mathematics and Applied Mathematics, Faculty of Science, Nelson Mandela University, PO Box 77000, Port Elizabeth 6031, South Africa

2

Department of Mathematics, Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

3

Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa

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The permutation character χ(L 3 (4):3|29 ) for the action of L 3 (4):3 on 29 will be computed and then together with the technique of coset analysis (see, for example, [5–8]), the conjugacy classes of G are computed. The inertia factor groups Hi for the action of L 3 (4):3 on Irr(29 ) and their fusion maps into L 3 (4):3 will also be determined. Next, the Fischer–Clifford matrices [9] of G and the ordinary character table of G associated with these matrices are computed. Finally, we will use the technique of set intersections (see, for example, [7,8,10]) to fully determined the fusion of the classes of G into U6 (2):3. This article, is part of a series of papers on the character tables of the maximal subgroups (29 :L 3 (4)):2 [11], (29 :L 3 (4)):3 and (29 :L 3 (4)):S3 [12] of the automorphism groups U6 (2):2, U6 (2):3 and U6 (2):S3 of U6 (2), respectively. The character table of the maximal subgroup 29 :L 3 (4) of U6 (2) is already uploaded in the GAP [13] library. The computation of the character tables of (29 :L 3 (4)):2 and (29 :L 3 (4)):3 using Fisher-Clifford theory (see, for example, th