International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 8684742 | 15 pages | https://doi.org/10.1155/2019/8684742

The Projective Character Tables of a Solvable Group

Academic Editor: Adolfo Ballester-Bolinches
Received07 Jun 2019
Accepted17 Sep 2019
Published27 Nov 2019

Abstract

The Chevalley–Dickson simple group of Lie type over the Galois field and of order has a class of maximal subgroups of the form , where is a special 2-group with center . Since is normal in , the group can be constructed as a nonsplit extension group of the form . Two inertia factor groups, and , are obtained if acts on . In this paper, the author presents a method to compute all projective character tables of . These tables become very useful if one wants to construct the ordinary character table of by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

1. Introduction

The Chevalley–Dickson simple group of Lie type over the Galois field and of order has exactly eight conjugacy classes of maximal subgroups [1]. One of these classes of maximal subgroups of index 1365 in is a 2-local subgroup of the form , where is a special 2-group with center . Also, it can be mentioned that is a parabolic subgroup of .

The group is the normalizer [1] of an elementary abelian 2-group of order 16 in , where the generators of are 4 commuting involutions found in the class of involutions of . Hence, we can construct as an extension group of by . Using a permutation representation of degree 416 of found in [2], we can easily verify with the help of the computer algebra system GAP [3] that exists as a nonsplit extension group . In fact, from the character table of uploaded in the GAP library, we can deduce that is also normal in . Therefore, with the aid of GAP, it can be easily shown that the group is also isomorphic to a nonsplit extension of the form .

Since , the action of on splits N into two orbits of lengths 1 and 15. By Brauer’s theorem in [4], G also acts on Irr with two orbits of lengths 1 and 15 and the corresponding inertia factor subgroups are of the forms and . If one wants to construct the ordinary character table of by the technique of Fischer–Clifford matrices [5], the irreducible ordinary characters Irr of and either the ordinary irreducible characters Irr or a set of irreducible projective characters IrrProj of with associated nontrivial factor set are needed. Readers are referred to [6] on a survey of Fischer–Clifford theory.

From the uploaded character table in the GAP library, we found that has 42 conjugacy classes. Hence, by Fischer–Clifford theory,  =  = 42. Therefore, , where . Since , certainly a set of irreducible projective characters with nontrivial factor set of is required in the construction of Irr via Fischer–Clifford theory.

Using computations in GAP, the Schur multiplier of is identified as a group isomorphic to the elementary abelian 2-group . Therefore, the group can have up to 15 sets of distinct irreducible projective characters , , such that . In this paper, the author presents a method (based on the work done in [7, 8]) to compute all the distinct irreducible projective character tables of . The author will also show how to choose the appropriate set for if one wants to construct the set Irr using the technique of Fischer–Clifford matrices.

A proof is given of a result, which states that the number of irreducible projective characters of a finite group G associated with some factor set is always less or equal to the number of the ordinary irreducible characters of G. Based on the abovementioned proof, GAP codes are developed to find the number of irreducible projective characters of associated with each factor set in a cohomology class of . Interested readers are referred to [913] for definitions on concepts in ordinary and projective character theory. Computations are carried out with the aid of the computer algebra systems MAGMA [14] and GAP [3], and the notation of ATLAS [1] is mostly used.

2. Preliminary Results on Projective Characters

In this section, a brief overview of relevant projective character theory pertaining to our study is given. G will always denote a finite group. Also, a proof of a proposition is given which states that the number of irreducible projective characters of a finite group G associated with some factor set α is always less or equal to the number of the ordinary irreducible characters of G.

Definition 1. A function α: is called a factor set of G if for all .
The set of all equivalence classes of factor sets of G forms a finite abelian group, called the Schur multiplier, and is denoted by .

Definition 2. A projective representation of a group G of degree n over the complex numbers is a map , such that(i)(ii)Given , there exists such that The map α is called the factor set associated with P.
Let P be a projective representation of G with factor set α. Define for all . Then, κ is called a projective character of G. We say that κ is irreducible if P is, and κ has a factor set α, where α is the factor set of P.
Let denote the set of irreducible projective characters of G associated with the factor set α. An element is said to be α-regular if for all . It is well known that is α- if and only if for some or equivalently that is α-irregular if and only if for all . The number of irreducible projective characters with factor set α equals the number of α-regular classes of a group G. Projective characters also satisfy the usual orthogonality relations and have analogues to ordinary characters.
As we will see later in Section 4, Definition 3 and Remark 1 will play an important role in the computation of the irreducible projective characters of .

Definition 3. A group R is a representation group for G if there exists a homomorphism π from R onto G such that (i) and (ii) .

Remark 1. A covering group C for G will normally be a quotient of R by a subgroup B of A. If has order n, we sometimes refer to the covering group as a n-fold cover of G. Projective representations of G are found in the representation group R for all the equivalence classes of factor sets in . However, in an n-fold cover C of G, only the n equivalence classes which C covers will be represented [7].
The following proposition (see [15, 16]) is useful to determine the number of irreducible projective characters of a group G associated with a certain factor set α. It tells us also under which condition is strictly less than . In Section 4, GAP codes (based on Proposition 1) will be used to compute the number of all irreducible projective characters of found in a set .

Proposition 1. Let . G be a representation group of a finite group G, where denotes the Schur multiplier of G. Then, the number of irreducible characters Irr of R which lie over a linear character is less or equal to .

Proof. The number of irreducible characters Irr of R which lies over a linear character is given by . It is known that the quantity for each , and it is nonzero if x is a commutator in R. For any , we havewhere is the number of conjugacy classes of . The last equality follows because the irreducible characters of R with in their kernels are precisely those which contain the trivial character on the restriction to . Hence,  =  =  = . Furthermore, if there is a nonidentity element which is a commutator in R, then the inequality becomes strict.

3. Action of on and

As explained in the introductory section, the split extension can be constructed as a nonsplit extension group of an elementary abelian group by . Since is the normalizer of N in , the action of on N gives rise to two orbits of lengths 1 and 15. It follows that the action of G on N will also result in two orbits of lengths 1 and 15 with corresponding point stabilizers and . Then, by Brauer’s theorem [4], G also has two orbits of lengths 1 and 15 on Irr with corresponding inertia factor subgroups and . See Table 1 for a summary of the action (which is self-dual) of on and , respectively.


Action of G on NAction of G on

Number and size of orbits


Structure of stabilizers and


Size of stabilizers and


Number of conjugacy classes of and


4. Projective Character Tables of Inertia Factor H2

In this section, all the sets IrrProj of irreducible projective characters of with associated factor sets will be computed. Appropriate GAP codes will be used to assist in the computational aspects of determining the sets IrrProj. Readers are referred to [7, 8] for a background on the computational techniques being used in this section.

Since the Schur multiplier of is isomorphic to the elementary abelian group of order 16, we obtain that contains 15 cohomology classes of order 2 and the trivial class . Hence, there exist 15 sets of projective characters IrrProj lying above the nontrivial factor sets , , such that . Note that the order of is the product of powers of two primes, and hence, by the Burnside’s theorem [17], the group is solvable. Since is of relative small order and it is solvable, the following GAP codes [18] (based on Proposition 1) give the number of irreducible projective characters of contained in each set IrrProj:gap h := gap f := EpimorphismSchurCover(h)gap f := InverseGeneralMapping(IsomorphismPcGroup(Source(f))) fgap z := Kernel(f)gap x := Source(f)gap  List(Irr(z), lambda Number(Irr(x), chi not IsZero(ScalarProduct(RestrictedClassFunction(chi,z), lambda))))

From the output of the above GAP codes, the number of irreducible projective characters of contained in each set is 30, 26, 18, 16, 16, 16, 16, 16, 16, 14, 4, 4, 4, 4, 4, and 4, respectively. The set containing the 30 projective characters is associated with the trivial factor set of , and hence, they are the ordinary irreducible characters Irr of . Note in the above GAP codes that “Source” is the full representation group of whereas “Kernel” denotes the Schur multiplier .

From Remark 1, the irreducible projective characters of with a given factor set can be computed without using the full representation group of . Since all the nontrivial factor sets of IrrProj are of order two, the aim is to find 15 double covers of which contain the desired 15 sets IrrProj of projective characters of . The nontrivial maximal subgroups of are computed within GAP, and we found there are exactly 15 of them of order 8 and shape . The 15 factor groups , , are the double covers which contain the 15 sets IrrProj of desired irreducible projective characters of with factor sets such that . The following GAP code was used to “restrict” R to to obtain the sets , :gapt:= CharacterTable(“”)gap2t := CharacterTable(“”)gapF := GetFusionMap(2t, t)gapmap := ProjectionMap(F)gapprojchars := List(2t, x{xmap})

From the sets which contain 16 irreducible projective characters each, we obtain exactly 3 distinct irreducible projective character tables. Also, the sets which contain 4 irreducible projective characters each, give rise to 3 distinct sets of irreducible projective character tables. Hence, we obtain 9 distinct sets of irreducible projective characters for , and they are listed in Tables 210. The ordinary irreducible characters Irr of are found in Table 11. Note that the irreducible projective characters with associated factor set have values of zero on the -irregular classes of G. Hence, the number of irreducible projective characters equals the number of -regular classes of G in each of the below tables. The above results are summarized in the following theorem:


1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

404000000011BB00DD400011BB00DD
404000000011BB00DD400011BB00DD
4040000000A/ABB00E−/E4000A/ABB00E−/E
4040000000/AABB00−/EE4000/AABB00−/EE
4040000000A/ABB00E/E4000A/ABB00E/E
4040000000/AABB00/EE4000/AABB00/EE
404000000011BB00DD−4000−1−1BB00DD
404000000011BB00DD−4000−1−1BB00DD
4040000000A/ABB00E/E−4000A−/ABB00E−/E
4040000000/AABB00/EE−4000−/AABB00−/EE
4040000000A/ABB00E−/E−4000A−/ABB00E/E
4040000000/AABB00−/EE−4000−/AABB00/EE
120−4000000000CB00000−40000CB0000
120−4000000000CB00000−40000CB0000
120−4000000000CB0000040000CB0000
120−4000000000CB0000040000CB0000

Note. A = ; B = −2i; C = −6i; D = i; E = .

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

4400000000112200−1−1−2−20011−4000−1−1
4400000000112200−1−12200−1−1400011
440000000011−2−200112200−1−1−4000−1−1
440000000011−2−20011−2−20011400011
4400000000A/A2200A−/A−2−200A/A−4000A−/A
4400000000/AA2200−/AA−2−200/AA−4000−/AA
4400000000A/A2200A−/A2200A−/A4000A/A
4400000000/AA2200−/AA2200−/AA4000/AA
4400000000A/A−2−200A/A2200A−/A−4000A−/A
4400000000/AA−2−200/AA2200−/AA−4000−/AA
4400000000A/A−2−200A/A−2−200A/A4000A/A
4400000000/AA−2−200/AA−2−200/AA4000/AA
12−40000000000−620000−620000040000
12−40000000000−6200006−200000−40000
12−400000000006−20000−6200000−40000
12−400000000006−200006−20000040000

Note. A = .

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

400040000011B000CC2200−1−1EE00CC
400040000011B000CC2200−1−1EE00CC
4000400000A/AB000D−/D2200A−/AEE00D/D
4000400000/AAB000−/DD2200−/AAEE00/DD
4000400000A/AB000D/D2200A−/AEE00D−/D
4000400000/AAB000/DD2200−/AAEE00−/DD
400040000011B000CC−2−20011EE00CC
400040000011B000CC−2−20011EE00CC
4000400000A/AB000D−/D−2−200A/AEE00D−/D
4000400000/AAB000−/DD−2−200/AAEE00−/DD
4000400000A/AB000D/D−2−200A/AEE00D/D
4000400000/AAB000/DD−2−200/AAEE00/DD
12000−400000000B0000−620000FE0000
12000−400000000B0000−620000FE0000
12000−400000000B00006−20000FE0000
12000−400000000B00006−20000FE0000

Note. A = ; B = 4i; C = i; D = ; E = 2i; F = 6i.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

808000000022000000000000000000
8080000000A/A000000000000000000
8080000000/AA000000000000000000
240-8000000000000000000000000000

Note. A = −1 − i3.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

880000000022000000000000000000
8800000000A/A000000000000000000
8800000000/AA000000000000000000
24-80000000000000000000000000000

Note. A = −1 − i3.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

800080000022000000000000000000
8000800000A/A000000000000000000
8000800000/AA000000000000000000
24000-80000000000000000000000000

Note. A = −1 − i3.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

4444440000−2−2000000000000000000
4444440000A/A000000000000000000
4444440000/AA000000000000000000
6−2−2−2−2600000000−2−20000BB0000BB00
6−2−2−2−2600000000−2−20000BB0000BB00
6−2−2−2−2600000000220000BB0000BB00
6−2−2−2−2600000000220000BB0000BB00
6−2−26−2−200000000−220000BB0000BB00
6−2−26−2−200000000−220000BB0000BB00
6−2−26−2−2000000002−20000BB0000BB00
6−2−26−2−2000000002−20000BB0000BB00
12−412−4−4−4000000000000000000000000
1212−4−4−4−4000000000000000000000000
12−4−4−412−4000000000000000000000000

Note. A =1 + i3; B = −2i.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

222222222222000000000000000000
2222222222A/A000000000000000000
2222222222/AA000000000000000000
666666−2−2−2−200000000000000000000
6−26−2−2−2−22−2200000000000−400000000
6−26−2−2−2−22−2200000000000400000000
6−26−2−2−22−22−20000000000−4000000000
6−26−2−2−22−22−200000000004000000000
66−2−2−2−2−2−22200000000000000000−400
66−2−2−2−2−2−22200000000000000000400
66−2−2−2−222−2−20000000000000000−4000
66−2−2−2−222−2−200000000000000004000
6−2−2−26−2−222−200000B00000000000000
6−2−2−26−2−222−200000B00000000000000
6−2−2−26−22−2−220000B000000000000000
6−2−2−26−22−2−220000B000000000000000
12−4−412−4−4000000000000000000000000
12−4−4−4−412000000000000000000000000

Note. A = −1 − i3; B = −4i.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

2222220000−1−1−2−200112200−1−1−2−20011
2222220000−1−1−2−20011−2−200112200−1−1
2222220000−1−12200−1−1−2−20011−2−20011
2222220000−1−12200−1−12200−1−12200−1−1
2222220000A/A−2−200A−/A2200A/A−2−200A−/A
2222220000/AA−2−200−/AA2200/AA−2−200−/AA
2222220000A/A−2−200A−/A−2−200A−/A2200A/A
2222220000/AA−2−200−/AA−2−200−/AA2200/AA
2222220000A/A2200A/A−2−200A−/A−2−200A−/A
2222220000/AA2200/AA−2−200−/AA−2−200−/AA
2222220000A/A2200A/A2200A/A2200A/A
2222220000/AA2200/AA2200/AA2200/AA
6−2−2−26−2000000−62000000BB0000BB00
6−2−2−26−2000000−62000000BB0000BB00
6−2−2−26−20000006−2000000BB0000BB00
6−2−2−26−20000006−2000000BB0000BB00
66−2−2−2−200000000BB0000BB006−20000
66−2−2−2−200000000BB0000BB006−20000
66−2−2−2−200000000BB0000BB00−620000
66−2−2−2−200000000BB0000BB00−620000
6−26−2−2−200000000BB00−62000000BB00
6−26−2−2−200000000BB00−62000000BB00
6−26−2−2−200000000BB006−2000000BB00
6−26−2−2−200000000BB006−2000000BB00
12−4−412−4−4000000000000000000000000
12−4−4−4−412000000000000000000000000

Note. A = ; B = −2i.

1a2a2b2c2d2e2f2g2h2i3a3b2j2k4a4b6a6b2l2m4c4d6c6d2n2o4e4f6e6f

768256256256256256646464641212192643232121219264323212121926432321212

111111111111111111111111111111
111111111111−1−1−1−1−1−1−1−1−1−1−1−1111111
111111111111−1−1−1−1−1−1111111−1−1−1−1−1−1
111111111111111111−1−1−1−1−1−1−1−1−1−1−1−1
1111111111A/A−1−1−1−1A−/A−1−1−1−1A−/A1111A/A
1111111111/AA−1−1−1−1−/AA−1−1−1−1−/AA1111/AA
1111111111A/A−1−1−1−1A−/A1111A/A−1−1−1−1A−/A
1111111111/AA−1−1−1−1−/AA1111/AA−1−1−1−1−/AA
1111111111A/A1111A/A−1−1−1−1A−/A−1−1−1−1A−/A
1111111111/AA1111/AA−1−1−1−1−/AA−1−1−1−1−/AA
1111111111A/A1111A/A1111A/A1111A/A
1111111111/AA1111/AA1111/AA1111/AA
333333−1−1−1−100−3−31100−3−3110033−1−100
333333−1−1−1−100−3−3110033−1−100−3−31100
333333−1−1−1−10033−1−100−3−31100−3−31100
333333−1−1−1−10033−1−10033−1−10033−1−100
6−26−2−2−2−22−2200000000−622−200000000
6−26−2−2−2−22−22000000006−2−2200000000
6−26−2−2−22−22−200000000−62−2200000000
6−26−2−2−22−22−2000000006−22−200000000
66−2−2−2−2−2−22200000000000000−622−200
66−2−2−2−2−2−222000000000000006−2−2200
66−2−2−2−222−2−200000000000000−62−2200
66−2−2−2−222−2−2000000000000006−22−200
6−2−2−26−2−222−200−622−200000000000000
6−2−2−26−2−222−2006−2−2200000000000000
6−2−2−26−22−2−2200−62−2200000000000000
6−2−2−26−22−2−22006−22−200000000000000
12−4−412−4−4000000000000000000000000
12−4−4−4−412000000000000000000000000

Note. A = .

Theorem 1. The inertia factor subgroup of the action of G on has exactly 9 distinct sets of with associated factor sets such that .

As mentioned in the introductory section, a set of 16 irreducible projective characters of with associated factor set is required in the construction of Irr, using the technique of Fischer–Clifford matrices. But a choice is to be made amongst the 3 distinct projective characters tables of which contain 16 irreducible projective characters each. The desired set of 16 irreducible projective characters can be found in the inertia group of on Irr.

Since we have a known permutation representation of , the group is generated within GAP. The normal subgroups of order 8 of are computed. There are three such groups, , and , which are elementary abelian. The 3 quotient groups are double covers for with each containing 46 ordinary irreducible characters. 30 characters from each of the sets Irr are the liftings of the ordinary irreducible characters of to , while the remaining 16 ordinary characters of each represent the set of irreducible projective characters of , which is of interest to us. Using the above GAP code as earlier, the desired set of projective characters of which is needed in the construction of the ordinary character table of using Fischer–Clifford theory is identified as Table 3.

5. Fischer–Clifford Matrices and Conjugacy Classes of

For the readers who are interested in the construction of the ordinary character table of using the technique of Fischer–Clifford matrices (see, for example, [6, 9, 1923]), the Fischer–Clifford matrices (Table 12) and the conjugacy classes (Table 13) of are supplied. The conjugacy classes are arranged in a format obtained by the technique of coset analysis (see, for example, [6, 9, 21, 24]). The set of irreducible characters of will be partitioned into 2 blocks and corresponding to the inertia factor groups and , respectively, where .