International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 9382525, 17 pages

https://doi.org/10.1155/2019/9382525

## Fischer-Clifford Matrices and Character Table of the Maximal Subgroup of

^{1}Department of Mathematics and Physics, Faculty of Applied Sciences, Cape Peninsula University of Technology, P.O. Box 1906, Bellville 7535, South Africa^{2}Department of Mathematics, Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

Correspondence should be addressed to Abraham Love Prins; az.ca.tupc@basnirp

Received 11 October 2018; Accepted 26 December 2018; Published 25 February 2019

Academic Editor: Adolfo Ballester-Bolinches

Copyright © 2019 Abraham Love Prins and Ramotjaki Lucky Monaledi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The automorphism group of the unitary group has a maximal subgroup of the form of order 20643840. In this paper, Fischer-Clifford theory is applied to the split extension group to construct its character table. Also, class fusion from into the parent group is determined.

#### 1. Introduction

The unitary simple group has outer automorphisms of orders 2, 3, and 6 and hence automorphism groups of the forms , , and exist for (see the ATLAS [1]). The reader is referred to [2] for more information about the construction of matrix representations for the covering and automorphism groups of . Recently in [3], a 3-local identification is given for the group and its automorphism groups , , and . Also, we found in the ATLAS that one of the 16 maximal subgroups of is a split extension group of the type of index 891 and has order 10321920. The group has automorphism groups of the form , , and which sit maximally inside the groups , , and , respectively.

The character table of is stored in the GAP Library [4], whereas the character table of and are not yet uploaded in GAP. In this paper, the unknown Fischer-Clifford matrices [5] of and its associated character table will be constructed. Readers are referred to [6] on a survey of Fischer-Clifford theory. It is interesting to note that our group is the stabilizer of a singular vector , where is the irreducible module of dimension 20 over the field for (see [3]). The method of coset analysis as discussed in [7–9] will be used in the computation of the conjugacy classes of elements of the group . The Fischer-Clifford matrices and character table of were determined in [10].

#### 2. Group

In this section, using a suitable permutation representation of , we identify our group as a split extension by with the aid of GAP [4] and MAGMA [11]. Then with the help of MAGMA we represent as a matrix group of degree 9 over the Galois field . Since acts absolutely irreducibly on its natural module , a split extension of the form exists. Then we create as a subgroup of and show with the help of MAGMA that is indeed an isomorphic copy of .

We construct within GAP, using its smallest permutation representation of degree 672 found in the online ATLAS of Finite Group Representations [12]. Next, we use the GAP commands “MS≔ ConjugacyClassesMaximalSubgroups ()”, “A1≔MS”, and “Size(A1)” to represent as a permutation group on 672 points and then use this permutation representation to construct within MAGMA. Using the sequence of MAGMA commands “a,b≔ChiefSeries(A1)”, “N≔ ”, “NormalSubgroups(A1)”, “IsNormal(A1,a)”, “IsElementaryAbelian(N)”, “C≔ Complements(A1,N)”, “Order(C)”, “C meet N”, and “IsIsomorphic(C, )”, we verified that .

Having as a permutation group on 672 points, we use the MAGMA commands “M≔GModule(A1,N)” and "‘M:Maximal” to represent as a matrix group of degree 9 over the Galois field . Thus we obtain the matrix group having 14 conjugacy classes of elements, where and . The generators and of are as follows:

The MAGMA command “IsAbsolutelyIrreducible(G)” tells us that the action of the matrix group on its natural module is absolutely irreducible. Thus a split extension of the type does exist. Hence we can construct as a subgroup of such that and , where , , and . The generators of the matrix group of degree 10 over are as follows:

Since we can represent and as a matrix and permutation group, respectively, we use the MAGMA command “IsIsomorphic ()’’ to confirm that . Hence we can regard as the split extension .

#### 3. The Conjugacy Classes of

In this section, the conjugacy classes of are computed using the technique of coset analysis and readers are encouraged to consult [8, 9] for a sound theoretical background on this technique.

Throughout the remainder of this chapter, let be a split extension of by , where is the vector space of dimension 9 over on which the linear group acts. Since is represented as a matrix group, we used the MAGMA commands “O≔ Orbits()”, “O”, “O”, “O”,O”, and “O” to compute the orbit lengths of the action of on . We obtain 4 orbits of lengths 1, 21, 210, and 280 and using the MAGMA commands “P1≔ Stabilizer(,O)”, “P2≔ Stabilizer(,O)”, “P3≔ Stabilizer(,O)”, and “P4≔ Stabilizer(,O)”, we are able to compute the corresponding point stabilizers , , which are subgroups of . With the aid of MAGMA and also checking the indices of the maximal subgroups of in the ATLAS, the structures of the stabilizers are identified as , , , and , where and are maximal subgroups of . We should note here that the group has two nonconjugate isomorphic maximal subgroups and , having the same structure . The stabilizer sits maximally in . Alternatively, we can use [13] to identify the structures of the groups . Since the action of on does not fix any nontrivial subspace of , we have that is an irreducible module for . We can readily verify this fact by using the MAGMA command “IsIrreducible()”.

Let be the permutation character of on the classes of . Then, from methods that were developed by Mpono [14], we obtain that = +++ = , where , , , and are the identity characters of the point stabilizers , , induced to . Note that the identity characters are identified with the permutation characters of acting on the classes of the point stabilizers . We found that = , = , = , and = . The permutation characters are written in terms of the ordinary irreducible characters of . Since we have the generators and for , we compute the character tables of and the ’s directly in MAGMA and use these tables together with the fusion maps of the stabilizers into , to compute and . The values of on the different classes of determine the number of fixed points of each in . The values of are listed in Table 1.