London Mathematical Society ISSN 1461–1570

LOW-DIMENSIONAL REPRESENTATIONS OF QUASI-SIMPLE GROUPS

GERHARD HISS and GUNTER MALLE

Abstract

We determine all absolutely irreducible representations of degree up to 250 of quasi-simple finite groups, excluding groups of Lie type in their defining characteristic. Additional information on the Frobenius-Schur indicators and the Brauer character fields of the representations is also given.

1. Introduction

In this paper we classify low-dimensional absolutely irreducible representations of quasisimple groups. Results of this type seem to have many applications. We have at least one immediate application in mind, namely to the matrix group recognition program in computational group theory.

Our main result is a list of all absolutely irreducible representations of quasi-simple groups up to degree 250, excluding representations of groups of Lie type in their defining characteristic, together with some information on these representations, like character field and Frobenius-Schur indicator.

Theorem 1. Let G be a quasi-simple finite group and V an absolutely irreducible faithful F G-module of dimension d ⩽ 250, where the characteristic of F is not the defining characteristic of G if G is of Lie type. Then (G,dim(V)) (and some additional information) are contained in Tables 1 and 2.

Previous work of Kondratiev (see [37] and the references therein) solved this question for representations up to degree 27. His results are recovered as part of our main theorem. In the case not considered here, of irreducible representations of groups of Lie type in their defining characteristic, Aschbacher (unpublished) obtained a classification up to degree 27. The recent work of Lübeck [38] determines all low-dimensional representations of groups of Lie type in their defining characteristic, thus complementing our list. More precisely, Lübeck determines all absolutely irreducible representations of G of degree at least up to l^{3}/8 where l is the Lie rank of G, including all those of degree at most 250.

The construction of the list naturally falls into three parts, according to the classification of finite simple groups. If the group G is of Lie type, we make use of results by various authors on low-dimensional representations in cross-characteristic (see [16, 17, 24] and the Landazuri-Seitz-Zalesskii bounds [49]. For the alternating groups and their spin-covers, results by James [28, 29] and Wales [55] can be used. Finally, for the sporadic groups there exists a wealth of published and unpublished tables of Brauer characters computed by different authors. In all three classes, the tables for the smaller groups can be found in the ordinary and modular Atlas [5, 33] and also in the computer algebra system GAP [14]. Still

Received . 1991 Mathematics Subject Classification 20C20, 20C40 \begin{matrix}c \\ ◯ \end{matrix} ????, Gerhard Hiss and Gunter Malle

LMS J. Comput. Math. ?? (????) 1–26

Low-dimensional representations

it turns out that in all three families there remain some groups where neither the theoretical results apply to determine all candidate representations, nor are the corresponding tables contained in GAP. In these cases the open questions could be solved with the help of the computer system MOC [32].

Throughout our paper we adopt the notation of the Atlas [5] for the quasi-simple groups. Acknowledgements. We thank Thomas Breuer for his help with GAP, in particular for writing GAP programs to handle some of the larger sporadic groups. We also thank Klaus Lux for his collaboration in the determination of the 7-modular character table of 2.𝔖_{14}, and Jürgen Müller for his assistance in computing some 2-modular characters of HN and J_{4}. Jürgen Müller has also written GAP programs to generate the character tables of 2.𝔄_{n} and constructed the 64-dimensional representation of 𝔄_{16} in characteristic 2 and the 128dimensional representation of 2.𝔄_{18} in characteristic 3. We thank Rod Gow for comments and references concerning the determination of Frobenius-Schur indicators, and Rob Wilson for the explicit computation of some of the more difficult indicators.

2. Groups of Lie type

In this section we describe the proof of correctness of the table for groups of Lie-type in non-defining characteristic.

2.1. Known results

The starting point is the result of Landazuri-Seitz-Zalesskii [49] giving lower bounds for degrees of non-trivial representations of quasi-simple groups of Lie type in non-defining characteristic. Using this we get down to the following finite list of simple candidate groups whose covering groups possibly have non-trivial irreducible representations of degree at most 250: L_{2}(q) (q ⩽ 499), L_{3}(q) (q ⩽ 13), L_{4}(q) (q ⩽ 5), L_{5}(2), L_{5}(3), L_{6}(2), L_{7}(2), U_{3}(q) (q ⩽ 16), U_{4}(q) (q ⩽ 5), U_{5}(q) (q ⩽ 4), U_{6}(2), U_{6}(3), U_{7}(2), U_{8}(2), U_{9}(2), S_{4}(q) (q ⩽ 19), S_{6}(q) (q ⩽ 7), S_{8}(q) (q ⩽ 3), S_{10}(q) (q ⩽ 3), O_{7}(3), O_{8}^{+} (2), O_{8}^{−} (2), O_{8}^{−} (3), 2B_{2}(8),^{2}B_{2}(32), G_{2}(q) (q ⩽ 5),^{3}D_{4}(2),^{3}D_{4}(3),^{2}F_{4}(2)′, F_{4}(2).

From this list we can eliminate the groups whose modular character tables are contained in the Atlas of Brauer Characters [33] or in GAP [14]. Furthermore, the irreducible representations in non-defining characteristic of L_{2}(q) are well-known (see [3]), and the relevant information is displayed in Table 1. For linear groups, Guralnick and Tiep [17] have described the representations of the first two non-trivial degrees and have given a lower bound for the third smallest degree. Similarly this has been done for unitary groups by the present authors in [24], and for the symplectic groups over fields of odd order in [16]. Using these results we are left with the groups L_{4}(4), L_{4}(5), U_{4}(4), S_{4}(7), S_{4}(8), S_{10}(2), O_{8}^{−} (3),^{3}D_{4}(3) and F_{4}(2).

Restriction to 2.O_{7}(3) (which is in GAP) shows that 2.O_{8}^{−} (3) has no faithful characters of degree at most 250, and the characters of the simple group O_{8}^{−} (3) are in GAP.

2.2. Some groups of Lie type and MOC

Here we comment on the computation of the results for some of the smaller groups of Lie type not contained in the Modular Atlas. For the linear groups L_{4}(4) and L_{4}(5) we used the modular representation theory of the general linear groups in the non-defining characteristic case as developed by Dipper and James in a series of papers. In particular we consulted reference [30], in which James computed the decomposition matrices for GL_{n}(q) up to n = 10.

For the unitary group U_{4}(4) we used the information contained in the diploma thesis of Wings [61], as well as the generic character table of U_{4}(q) computed by Nozawa [44]. Partial decomposition matrices for the 4-dimensional symplectic groups have been obtained by White, and completed by Okuyama and Waki in [45]. For the case of cyclic defect groups, the decomposition matrices have been computed by Fong and Srinivasan [7]. To deal with the groups S_{4}(7) and S_{4}(8) we used White’s papers [56], [57] and [58]. For the character values we consulted the generic character tables (see [50], [46] and [6]). The desired result for ^{3}D_{4}(3) can be extracted from Geck’s work [8], and the one for F_{4}(2) is contained in [21].

The decomposition matrices for F_{4}(2) as well as those for S_{10}(2) and many sporadic groups have been computed with the help of MOC [32], a system for dealing with modular characters. This was developed by Jansen, Lux, Parker and the first author beginning in 1984. A description of MOC and its algorithms is given in [32], which unfortunately never found a publisher. Some of the algorithms and methods used are also described in [39, Section 3] and in [40, Section 3].

Let us briefly sketch the principal idea without going into too many details. Let G be a finite group and let p be a prime dividing the order of G. Suppose that the ordinary character tables of G and the p-modular character tables of some (maximal) subgroups of G are known. In a first step we compute a large set B of p-modular Brauer characters and a large set P of projective characters of G. This is done with the usual methods, by tensoring characters or by inducing them from subgroups.

Let IBr(G) and IPr(G) denote the set of irreducible Brauer characters and of projective indecomposable characters of G, respectively. In a second step we compute a basic set of Brauer characters BS from B and a basic set of projective characters PS from P. The term basic set here means that the transition matrices X and Y with BS = X · IBr(G) and PS = Y · IPr(G) are unimodular and have non-negative integral entries. (In the above matrix equations we have identified the sets of characters with their matrices of values, so that, for example, IBr(G) also stands for the Brauer character table of G.)

Let U denote the square matrix of scalar products between the characters in BS and those in PS. Then U = XY^{t}, by the orthogonality relations. Thus in order to find candidates for IBr(G) one has to find the factorizations of U into a product of unimodular matrices with non-negative integral entries. If U = X_{1}Y_{1}^{t} is such a factorization, then X_{1}^{−1} ·BS is a candidate set for the irreducible Brauer characters. The projective characters in P reduce the number of candidates for IBr(G), and thus for the first factor X_{1}, since every irreducible Brauer character must have a non-negative scalar product with every character in P. Dually, the Brauer characters in B restrict the number of possibilities for the second factor Y_{1}^{t}.

Of course, in each individual problem we used some ad hoc techniques, adapted to the particular problem. The application of these techniques was supported by substantial calculations with GAP and MAPLE [4].

3. The alternating groups

In this section we collect some results on low-dimensional representations of alternating groups and their covering groups. For this purpose we recall some standard notation. For a field F and a partition λ ⊢ n let S^{λ} denote the corresponding Specht module for 𝔖_{n} over F. Assume that F has characteristic ℓ. For ℓ-regular partitions λ we write D^{λ} for the irreducible F 𝔖_{n}-module indexed by λ, the unique simple composition factor in the socle of S^{λ}. It is known that the D^{λ} constitute a complete set of representatives for the irreducible F 𝔖_{n}-modules.

3.1. 2-modular representations of 𝔄_{n}

We first consider the case of representation over fields of characteristic ℓ = 2. The 2modular character table of 𝔖_{n} is known for n ⩽ 14 [33, 1].

Proposition 2. Let n ⩾ 15, ℓ = 2 and (n) ≠ λ ⊢ n be 2-regular. Assume that dim(D^{λ}) ⩽ 500. Then one of the following cases occurs.

(1) λ = (n − 1,1) and n ⩽ 502, or

(2) λ = (n − 2,2) and n ⩽ 34, or

(3) λ = (12,3), dim(D^{λ}) = 336, or

(4) λ = (8,7),(9,7), dim(D^{λ}) = 128, or

(5) λ = (9,8),(10,8), dim(D^{λ}) = 256.

Proof. From the known 2-modular character table of 𝔖_{14} it follows that for n = 14 either

λ ∈ M_{14} := {(14),(13,1),(12,2),(11,3),(10,4),(8,6)}

or dim(D^{λ}) > 500. Now let n = 15 and assume that dim(D^{λ}) ⩽ 500. Then the restriction D^{λ}|_{𝔖_{14}} has to contain some D^{μ} with μ ∈ M_{14} as a submodule. This implies that S^{λ} is a constituent of the induced module (S^{μ})^{𝔖_{15}}, hence

λ ∈ {(15),(14,1),...,(9,6),(8,7),(8,6,1)}

by the branching rule for ordinary characters of 𝔖_{n}. The dimension of D^{λ} for λ in the above list is known by [28, Th. 5.2 and Th. 7.1], and it turns out that dim(D^{λ}) ⩽ 500 only for

λ ∈ M_{15} := {(15),(14,1),(13,2),(12,3),(8,7)}.

The same argument may be applied to obtain M_{16},...,M_{19} yielding in particular M_{19} = {(19),(18,1),(17,2)}. Now induction and the branching rule allow to complete the proof. The degrees of the relevant D^{λ} follow from [28, Th. 5.2].

Benson has determined those 2-regular partitions λ = (λ_{1} ⩾ λ_{2} ⩾ ...) for which the irreducible 𝔖_{n}-module D^{λ} splits upon restriction to the alternating group 𝔄_{n}: the restriction is irreducible if λ_{2j−1} − λ_{2j} > 2 or λ_{2j−1} + λ_{2j} ≡ 2 (mod4) for some j > 0 [2, Th. 1.1]. We thus obtain:

Corollary 3. Let n ⩾ 15, ℓ = 2 and D a non-trivial absolutely irreducible 𝔄_{n}-module of dimension dim(D) ⩽ 250. Then either D is the restriction to 𝔄_{n} of D^{(n−1,1)} with n ⩽ 252 or of D^{(n−2,2)} with n ⩽ 24, or

(n,dim(D)) ∈ {(15,64),(16,64),(17,128)}.

3.2. ℓ-modular representations of 𝔄_{n}, ℓ > 2

Now assume that F is a sufficiently large field of characteristic ℓ ≠ 2. The decomposition matrices of 𝔖_{n} over F are known for n ⩽ 14 (see [33]) or can be computed using the Specht-package in GAP written by Andrew Mathas. (In fact, for ℓ ≠ 3 the decomposition matrices can be computed at least up to n = 20, but we will not need this here.)

We obtain the following analogue of Proposition 2:

Proposition 4. Let n ⩾ 14 and let M be an non-trivial absolutely irreducible F 𝔄_{n}-module of dimension dim(M) ⩽ 250. Then M = D^{λ} for λ one of (1) (n − 1,1) and n ⩽ 252, or

(2) (n − 2,2) and n ⩽ 24, or

(3) (n − 2,1^{2}) and n ⩽ 24, or

(4) (11,3) and ℓ = 5,7, or

(5) (7,7) and ℓ = 5.

Proof. We can argue essentially as in the proof of the preceding proposition. First let ℓ = 3. From the known decomposition matrix it follows that the irreducible 𝔄_{13}-modules D with dim(D) ⩽ 250 are the D^{λ} with

λ ∈ M_{3,13} := {(13),(12,1),(11,2),(11,1^{2}),(10,3),(10,2,1)},

in particular, all of them are restrictions of irreducible 𝔖_{13}-modules. Thus if D is an irreducible 𝔄_{14}-module of dimension 1 ≠ dim(D) ⩽ 250 then D is a constituent of D^{λ} or of Dλ̄ for a partition λ ⊢ 14 which can be obtained by adding a hook of length 1 to some partition μ ∈ M_{3,13}. The dimensions of the ten relevant D^{λ} can be determined using [29]. It turns out that in fact

λ ∈ M_{3,14} := {(14),(13,1),(12,2),(12,1^{2})}.

All of these restrict irreducibly to 𝔄_{14}. Similarly for n = 15 we obtain D = D^{λ} for

λ ∈ M_{3,15} := {(15),(14,1),(13,2),(13,1^{2})}

and so on. The assertion for ℓ = 3 follows by induction.

The argument for ℓ ⩾ 5 is similar. For ℓ = 0 the degrees of the Specht modules are well known.

3.3. ℓ-modular representations of 2.𝔄_{n}

Here we study low-dimensional faithful irreducible representations of the covering groups 2.𝔄_{n}. Since an irreducible 2-modular representation of 2.𝔄_{n} is not faithful, we assume that ℓ ≠ 2 throughout this section. Also, we may and will assume that n ⩾ 14 since for smaller n the modular tables are contained in [33] or in GAP.

We begin by looking at the covering groups of the symmetric groups. Their irreducible representations in characteristic 0 are indexed by partitions of n into distinct parts, with the rule that one partition corresponds to two different representations if these have the same restriction to 2.𝔄_{n}. Following the notation in the paper of Wales [55], the representations of 2.𝔖_{n} corresponding to a partition λ are denoted by < λ >, respectively by < λ >, < λ >^{∗} if there are two.

Proposition 5. Let n ⩾ 14 and ℓ ≠ 13 for n = 14. Then a faithful irreducible F 2.𝔖_{n}-module M with dimM ⩽ 500 is a constituent of the reduction modulo ℓ of < n > or of < n >^{∗}.

Proof. The assertion is correct for n = 14 and ℓ ≠ 13 and for n = 15 and ℓ = 13. This follows from explicit computations with the character tables using MOC.

Suppose then that n ⩾ 15 and n ⩾ 16 if ℓ = 13. Let M be a faithful simple F 2.𝔖_{n+1} - module with dimM ⩽ 500. Then the restriction of M to 2.𝔖_{n} contains a faithful irreducible submodule of dimension at most 500. By induction, this is a composition factor of < n > or of < n >^{∗}.

It follows that M is a composition factor of the reduction modulo ℓ of < n >^{2.𝔖_{n+1}} or of < n >^{∗} ^{2.𝔖_{n+1}} . By the branching theorem we have < n >^{2.𝔖_{n+1}} =< n + 1 > + < n,1 > if n is odd, and < n >^{2.𝔖_{n+1}} =< n + 1 > + < n + 1 >^{∗} + < n,1 > if n is even. By Table IV of [55], a modular composition factor of < n,1 > either is a composition factor of < n + 1 > or < n + 1 >^{∗}, or its dimension is larger than 500. This completes the proof.

This proposition shows that for n ⩾ 15 and n = 14, ℓ ≠ 13, a faithful irreducible F 2.𝔄_{n}- module of dimension at most 250 is a composition factor of the restriction to 2.𝔄_{n} of the reduction modulo ℓ of < n > or of < n >^{∗}. A direct computation with MOC shows that this is true even for n = 14 and ℓ = 13. (Note that this is a cyclic defect case.) The modular composition factors of < n > and < n >^{∗} are determined in [55]. In addition, it is also determined there which of them restrict irreducibly to 2.𝔄_{n}.

We thus obtain from Table III of [55]:

Corollary 6. Let n ⩾ 14, ℓ ≠ 2 and let D be a faithful absolutely irreducible 2.𝔄_{n}-module of dimension dim(D) ⩽ 250. Then n ⩽ 18 and D is a composition factor of a basic spin representation of 2.𝔄_{n}. We have the following cases for (n,dim(D)):

(1) (14,32) for ℓ = 7 (two D’s),

(2) (14,64) for ℓ ≠ 7 (one D),

(3) (15,64) for ℓ = 3,5 (one D),

(4) (15,64) for ℓ ≠ 3,5 (two D’s),

(5) (16,128) (one D),

(6) (17,128) for ℓ = 17 (one D),

(7) (17,128) for ℓ ≠ 17 (two D’s),

(8) (18,128) for ℓ = 3 (two D’s).

4. The sporadic groups

Our first source was the Modular Atlas [33], which contains the modular character tables for ten of the 26 sporadic groups (and their covering groups). Apart from these ten groups, the largest of which is the McLaughlin group., the modular character tables are completely known for the six sporadic groups He, 2.Ru, 6.Suz, 3.O′N, Co_{3} and Co_{2}. Modular character tables for some of the larger groups are also available in some characteristics, for example for the Fischer groups 6.F i_{22} and F i_{23} in characteristic 5 or for the Conway group 2.Co_{1} in characteristic 7. Moreover, for primes dividing the group order only once, the character degrees can be computed from the Brauer trees given in [22].

There are several sources for these character tables, not all of them have been published yet. The only published source on the Held group is Ryba’s paper [47]. The 2-modular character tables for the covering group of the Rudvalis group were determined by Wilson (unpublished), its 3-modular tables are contained in [20] and its 5-modular tables in [25]. On the Suzuki group there is only one paper, by Jansen and Müller [34]. The relevant papers for the O’Nan group and its covering group are [35], by Jansen and Wilson, and [18], by Henke, Hiss and Müller. The 2-modular characters of the smallest two Conway groups have been computed by Suleiman and Wilson [51], [52], its 3-modular characters by Jansen in his PhD-thesis [31], and the 5-modular characters of Co_{3} by Müller [43]. The 5-modular character table for Co_{2} is contained in [32], as well as the 7-modular table for 2.Co_{1}. The 5-modular Brauer characters for the Fischer groups 6.F i_{22} and F i_{23} have been computed by Hiss, Lux, and White in [23] and [26]. This completes the list of known modular tables for the sporadic groups and their covering groups. The Web site of MOC [15] contains the corresponding decomposition matrices and Brauer character degrees.

The latter reference also contains a list of minimal degrees of non-trivial representations of sporadic groups and their covering groups (except for the double cover of the Baby Monster), compiled by Christoph Jansen. It follows from this list that we do not have to consider the two Fischer groups F i_{23}, F i′_{24}, nor the Baby Monster B or the Monster M. For example, the minimal degree of a non-trivial representation of F i_{23} is 253. Since F i_{23} is a subgroup of the double cover 2.B of the Baby Monster, the smallest degree of the latter group is well above 250.

The same list also shows that for the Lyons group Ly we only have to worry about representations in characteristic 5, and for the largest Janko group J_{4} only about characteristic 2. The case of the Lyons group was handled by Klaus Lux and Alex Ryba (unpublished). They showed, using the condensation techniques (see [48] or [41]) of the Meat-Axe, that the only non-trivial representation in characteristic 5 of the Lyons group of degree smaller than 250 is the known one of degree 111, constructed by Meyer, Neutsch and Parker [42]. Lux and Ryba also settled the case of the Harada-Norton group in characteristic 5 with the same techniques, but this has been checked independently with the methods described in Section 2. The minimal degree of a faithful representation of the Conway group in odd characteristic equals 276. Hence for 2.Co_{1} and odd characteristics we only have to consider blocks containing faithful characters. The diploma thesis of Hensing [19] gives enough information on the 5-modular decomposition numbers of 2.Co_{1} to show that our list is correct in this case. Unpublished work of Rosenboom settles the case of Co_{1} in characteristic 2, but we have checked this case independently. The case of the smallest Fischer group F i_{22} and its covering group has been handled by Christoph Jansen (unpublished).

We are thus left with the groups HN, T h, 2.Co_{1} and J_{4}, the latter only in characteristic 2, and the Conway group only in characteristic 3. To deal with these cases we used the methods and the MOC system described in Section 2.

5. The Frobenius-Schur indicators

We briefly comment on the methods used to determine the Frobenius-Schur indicators of the characters occurring in the tables below. Let us begin with the definition. Suppose that G is a finite group, K a field, and M a finitely generated, absolutely irreducible KGmodule. Suppose first that M is self-dual. Then M carries a non-degenerate G-invariant bilinear form, which is symmetric or alternating. Moreover, this form is unique up to scalar multiplication. If the characteristic of K is 2, and M is not the trivial module, then the invariant form is alternating. If the characteristic of K is odd and M carries a non-degenerate symmetric bilinear form, this form is the polarization of a quadratic form on M, which is non-degenerate and G-invariant. If M is not self-dual, it does not carry any non-trivial G-invariant bilinear form.

The Frobenius-Schur indicator of M is an integer of the set {−1,0,+1}. It is defined to be 0, if and only if M is not self-dual. The Frobenius-Schur indicator of M is set to be +1, if and only if M carries a non-degenerate G-invariant quadratic form. Thus the indicator of M is −1 if and only if M is self-dual and carries a non-degenerate G-invariant bilinear alternating form, but no G-invariant quadratic form.

In our tables the indicators are represented by one of the three symbols {−,∘,+} with the obvious bijection.

To compute the Frobenius-Schur indicators, we used several well known methods. We begin our discussion with the case that K has odd characteristic p and M is self-dual. Then the Frobenius-Schur indicator of M can be computed from the p-modular decomposition matrix of G and the ordinary character table of G as follows. There is a self-dual ordinary irreducible module V of G whose reduction modulo p contains M with odd multiplicity, and the Frobenius-Schur indicator of M is equal to that of V. This result is due, independently, to Thompson [53] and Willems [59] (see the survey [60], in particular Theorem 2.8 therein).

In our case, the self-dual ordinary irreducible modules containing M with odd multiplicity can easily be found; in fact most of the time M is liftable to a self-dual ordinary irreducible module. Thus it remains to compute the Frobenius-Schur indicators of certain ordinary irreducible modules.

Let V be an irreducible ℂG-module with character χ. Then the Frobenius-Schur indicator of V is equal to

ν_{2}(χ) := \frac{1}{|G|} ∑_{g∈G}χ(g^{2}).

Thus the indicators in characteristic 0 can be computed from the ordinary character table and the 2-powermap of G. Also, ν_{2}(χ) = +1 if and only if V ≅ ℂ ⊗_{ℝ} V_{0} for some ℝGmodule V_{0}, in other words, if and only if χ is afforded by a real representation of G (see Chapter 4 of [27]). If the character table of G is not available, we use this latter criterion.

Let χ be real valued. To decide whether or not χ is afforded by a real representation, we use the real Schur index m_{ℝ}(χ) (see Definition 10.1 of [27]). By the definition of the Schur index and the discussion above we have ν_{2}(χ) = −1 if and only if m_{ℝ}(χ) ≠ 1. Since |ℂ:ℝ| = 2, the latter condition is equivalent to m_{ℝ}(χ) = 2, and implies that χ(1) is even (see Corollary 10.2(g) and (h) of [27]). Suppose that ψ is the character of a real representation of G. Then, by Corollary 10.2(c) of [27], the Schur index m_{ℝ}(χ) divides the scalar product (ψ,χ). This can be applied in particular in case ψ is induced or restricted from a character of a real representation of a subgroup, respectively overgroup, of G.

Results on Schur indices of characters of groups of Lie type can be found in the papers [9, 10] by Rod Gow. We also made use of the results of Przygocki [46] on the Schur indices of the symplectic groups and of Tiep [54], Cor. 4.5, on the Schur indices of the Weil representations of the unitary groups.

Let us suppose now that the characteristic of K equals 2. Here, no simple method is known for computing the Frobenius-Schur indicators. A collection of various methods is contained in the articles [12, 13] of Gow and Willems. We made in particular use of Lemmas 1.1 and 1.2 of [12], and of Theorem 1.1 of [13] and its proof.

If the KG-module M lifts to an absolutely irreducible ℚG-module, then the Frobenius-Schur indicator of M equals +1. This is a well known elementary result which is included in the much more general Theorem 2.2 of [12].

To determine the indicators of the basic spin representations of the alternating groups, we followed a suggestion of Rod Gow. First, by Corollary 4.3 of [11], the spin module of the symplectic group S_{2m}(q), q even, of degree 2^{m} has indicator +1, if m ⩾ 3 (see also Proposition 5.4.9 of [36]). The restriction of the spin module of S_{2m}(q) to the subgroup 𝔖_{2m+1} or 𝔖_{2m+2} remains irreducible. Thus the basic spin representation of the symmetric group 𝔖_{n} has indicator +1 if n ⩾ 7. This implies, for example with Lemma 1.2 of [12], that the basic spin representations for 𝔄_{n} also have indicator +1, provided they are selfdual. The splitting of the basic spin representation of 𝔖_{n} on restriction to 𝔄_{n} has been determined by Benson in Theorems 1.1 and 6.1 of [2].

The Frobenius-Schur indicators for the sporadic groups have mainly been determined by Richard Parker and Rob Wilson using the computational methods described in [52]. In addition, Rob Wilson computed the indicators of the following representations

\begin{matrix}d & G & ℓ & ind \\ 32 & 2.𝔄_{14} & 7 & − \\ 104 & U_{4}(5) & 2 & − \\ 174 & S_{4}(7) & 2 & + \\ 208 & 𝔄_{14} & 2 & + \\ 218 & ^{3}D_{4}(3) & 2 & + \end{matrix}

which remained open in a previous version of this paper.

6. A table of low-dimensional absolutely irreducible representations of quasi-simple groups

In order to shorten the table of low-dimensional representations we have grouped together some generic cases in the following table. The corresponding entries have been omitted from the large table for 𝔄_{n} with n ⩾ 14 and for L_{2}(q) for q ≠ 9. The indicators for L_{2}(q) when ℓ = 2 were taken from [12, Th. 2.3] for the characters of degree q ± 1, and from [13, Th. 1.2] for the Weil character of degree (q − 1)/2.

It is maybe helpful to point out that in accordance with the formulation of Theorem 1 the following pairs (G/Z(G),ℓ) have been omitted from the tables:

𝔄_{6} ≅ S_{4}(2)′ ≅ L_{2}(9) in characteristic ℓ = 3,

𝔄_{8} ≅ L_{4}(2) in characteristic ℓ = 2,

L_{2}(7) ≅ L_{3}(2) in characteristic ℓ = 2,7,

U_{4}(2) ≅ S_{4}(3) in characteristic ℓ = 2,3,

G_{2}(2)′ ≅ U_{3}(3) in characteristic ℓ = 3,

2G_{2}(3)′ ≅ L_{2}(8) in characteristic ℓ = 2.

The Tits group, that is, the derived group ^{2}F_{4}(2)′of the Ree group of type F_{4} over 𝔽_{2}, cannot be obtained as a factor group of the group of fixed points under a Frobenius morphism of an algebraic group, so it is not considered as a group of Lie type in characteristic 2 here.

Let us briefly explain the notation used in the tables. The column headed “d” contains the degrees. The column headed “ℓ” specifies the characteristics of the fields over which the respective representations are defined. An entry ≠ ℓ for a prime ℓ indicates that a representation of this degree exists for all other characteristics than ℓ, including 0. The “field” column gives the irrationalities of the Brauer characters (using Atlas notation), except for the irrationalities occurring in Table 1. Finally, the last column gives the Frobenius-Schur indicators, which are defined in the section above.

Low-dimensional representations

Table 7. 1. Generic examples.

Alternating groups

d  G  ℓ field  ind
n − 2  𝔄_{n}  ℓ|n  +
n − 1  𝔄_{n}  ℓ |̸n  +

L_{2}(q), q ≡ 1 (mod4), ℓ |̸q

d  G  condition on ℓ field  ind
(q − 1)/2 L_{2}(q)  2 \sqrt{q}  −
(q − 1)/2 2.L_{2}(q)  ℓ ≠ 2 \sqrt{q}  −
(q + 1)/2 L_{2}(q)  ℓ ≠ 2 \sqrt{q}  +
q − 1  L_{2}(q)  ζ_{q+1}^{j} + ζ_{q+1}^{−j} +
q − 1  2.L_{2}(q)  ((q + 1)/2)_{ℓ′}≠ 1 ζ_{q+1}^{j} + ζ_{q+1}^{−j} −
q  L_{2}(q)  ℓ |̸(q + 1)  +
q + 1  L_{2}(q)  ((q − 1)/4)_{ℓ′}≠ 1 ζ_{q−1}^{2j} + ζ_{q−1}^{−2j} +
q + 1  2.L_{2}(q)  ℓ ≠ 2 ζ_{q−1}^{j} + ζ_{q−1}^{−j} −

L_{2}(q), q ≡ 3 (mod4), ℓ |̸q

d  G  condition on ℓ field  ind
(q − 1)/2 L_{2}(q)  \sqrt{−q}  ∘
(q + 1)/2 2.L_{2}(q)  ℓ ≠ 2 \sqrt{−q}  ∘
q − 1  L_{2}(q)  ζ_{q+1}^{2j} + ζ_{q+1}^{−2j} +
q − 1  2.L_{2}(q)  ℓ ≠ 2 ζ_{q+1}^{j} + ζ_{q+1}^{−j} −
q  L_{2}(q)  ℓ |̸(q + 1)  +
q + 1  L_{2}(q)  ((q − 1)/2)_{ℓ′}≠ 1 ζ_{q−1}^{j} + ζ_{q−1}^{−j} +
q + 1  2.L_{2}(q) ((q − 1)/2)_{ℓ′}≠ 1,ℓ ≠ 2 ζ_{q−1}^{j} + ζ_{q−1}^{−j} −

L_{2}(q), q ≡ 0 (mod2), ℓ |̸q

d  G  condition on ℓ field  ind
q − 1  L_{2}(q)  ζ_{q+1}^{j} + ζ_{q+1}^{−j} +
q  L_{2}(q)  ℓ |̸(q + 1)  +
q + 1  L_{2}(q)  (q − 1)_{ℓ′}≠ 1 ζ_{q−1}^{j} + ζ_{q−1}^{−j} +

Low-dimensional representations

Table 8. 2. Absolutely irreducible representations of quasi-simple groups.

d G  ℓ field  ind
3 3.𝔄_{6}  0,2 z3,b5 ∘
3 3.𝔄_{6} 5 z3 ∘
3 3.𝔄_{7}  5 z3,b7 ∘
4 𝔄_{6}  2  −
4 2.𝔄_{6}  0,5  −
4 𝔄_{7}  2 b7  ∘
4 2.𝔄_{7}  7  −
4 2.𝔄_{7}  ≠ 2,7 b7  ∘
4 4_{2}.L_{3}(4)  3 i1,r7  ∘
4 2.U_{4}(2) 0,5 z3  ∘
5 𝔄_{6}  0,5  +
5 𝔄_{7}  7  +
5 U_{4}(2)  0,5 z3  ∘
5 M_{11}  3 i2,b11 ∘
6 3.𝔄_{6}  0,5 z3  ∘
6 6.𝔄_{6}  0,5 z3,r2  ∘
6 𝔄_{7}  ≠ 7  +
6 2.𝔄_{7}  3 r2  −
6 3.𝔄_{7}  ≠ 3 z3  ∘
6 6.𝔄_{7}  ≠ 2,3 z3,r2  ∘
6 2.L_{3}(4)  3  +
6 6.L_{3}(4) ≠ 2,3 z3  ∘
6 U_{3}(3)  ≠ 3  −
6 U_{4}(2)  0,5  +
6 3_{1}.U_{4}(3)  2 z3  ∘
6 6_{1}.U_{4}(3) ≠ 2,3 z3  ∘
6 2.M_{12}  3 i2,i5,b11 ∘
6 3.M_{22}  2 z3,b11 ∘
6 J_{2}  2 b5  −
6 2.J_{2}  5  −
6 2.J_{2}  ≠ 2,5 b5  −
7 𝔄_{8}  ≠ 2  +
7 𝔄_{9}  3  +
7 U_{3}(3)  0,7  +
7 U_{3}(3)  0,7 i1  ∘
7 S_{6}(2)  ≠ 2  +
7 J_{1}  11 b5,c19 +
8 𝔄_{6}  0,2 b5  +
8 𝔄_{6}  5  +
8 2.𝔄_{6}  0 b5  −
8 𝔄_{7}  5  +
8 2.𝔄_{8}  ≠ 2  +
8 𝔄_{9}  ≠ 3  +
8 2.𝔄_{9}  ≠ 2  +
8 𝔄_{10}  2  −
8 𝔄_{10}  5  +

d G  ℓ field  ind
8 2.𝔄_{10}  5 r6,r21  +
8 4_{1}.L_{3}(4)  5 i1  ∘
8 4_{1}.L_{3}(4) ≠ 2,5 i1,b5  ∘
8 2.S_{6}(2) ≠ 2 +
8 2.O_{8}^{+} (2) ≠ 2  +
8 2.Sz(8)  5 c_{13}  +
9 𝔄_{6}  0  +
9 3.𝔄_{6}  0,2 z3  ∘
9 3.𝔄_{7}  7 z3  ∘
9 𝔄_{10}  ≠ 2,5  +
9 𝔄_{11}  11  +
9 M_{11}  11  +
9 3.J_{3}  2 z3,b17,b19 ∘
10 𝔄_{6}  0,5  +
10 2.𝔄_{6}  0,5 r2  −
10 𝔄_{7}  7  +
10 𝔄_{7}  ≠ 2,7 b7  ∘
10 𝔄_{11}  ≠ 11  +
10 𝔄_{12}  2,3  +
10 2.L_{3}(4)  7  +
10 2.L_{3}(4) ≠ 2,7 b7  ∘
10 U_{4}(2)  0,5 z3  ∘
10 U_{5}(2)  ≠ 2  −
10 M_{11}  ≠ 11  +
10 M_{11}  ≠ 2 i2  ∘
10 M_{12}  2,3  +
10 2.M_{12}  ≠ 2 i2  ∘
10 M_{22}  2 b7  ∘
10 2.M_{22}  7  +
10 2.M_{22}  ≠ 2,7 b7  ∘
11 𝔄_{12}  ≠ 2,3  +
11 𝔄_{13}  13  +
11 U_{5}(2)  ≠ 2,3 z3  ∘
11 M_{11}  ≠ 2,3  +
11 M_{12}  ≠ 2,3  +
11 M_{23}  2 b7,i15,b23 ∘
11 M_{24}  2 b7,i15,b23 ∘
12 6.𝔄_{6}  0 z3,b5  ∘
12 6.𝔄_{7}  5 z3,b7  ∘
12 𝔄_{13}  ≠ 13  +
12 12_{2}.L_{3}(4)  7 z12,b5  ∘
12 U_{3}(4)  ≠ 2  −
12 S_{4}(5)  2 b5  −
12 2.S_{4}(5) ≠ 2,5 b5  −
12 2.G_{2}(4)  ≠ 2  −
12 2.M_{12}  ≠ 2,3  +

d G  ℓ field  ind
12 2.Suz  3  −
12 3.Suz  2 z3  ∘
12 6.Suz  ≠ 2,3 z3  ∘
13 𝔄_{7}  3,5  +
13 𝔄_{8}  3,5  +
13 U_{3}(4)  ≠ 2,5 z5  ∘
13 S_{4}(5)  ≠ 2,5 b5  +
13 S_{6}(3)  ≠ 3 z3  ∘
13 J_{2}  3 b5  +
14 𝔄_{7}  ≠ 3,5  +
14 2.𝔄_{7}  ≠ 2,3 r2  −
14 𝔄_{8}  0,7  +
14 U_{3}(3)  ≠ 3  +
14 S_{6}(2)  3  +
14 2.S_{6}(3) ≠ 2,3 z3  ∘
14 Sz(8)  ≠ 2 i1  ∘
14 G_{2}(3)  ≠ 3  +
14 J_{1}  11 b5,c19  +
14 J_{2}  5  +
14 J_{2}  ≠ 3,5 b5  +
14 2.J_{2}  ≠ 2  −
15 3.𝔄_{6}  0,5 z3  ∘
15 𝔄_{7}  ≠ 2,7  +
15 3.𝔄_{7}  ≠ 3 z3  ∘
15 L_{3}(4)  3  +
15 3.L_{3}(4) ≠ 2,3 z3  ∘
15 U_{4}(2) 0,5 +
15 3_{1}.U_{4}(3) ≠ 3 z3  ∘
15 S_{6}(2)  ≠ 2,3  +
15 M_{12}  3 b11  ∘
15 3.M_{22}  2 z3,b11  ∘
16 2.𝔄_{7}  7  −
16 2.𝔄_{8}  7  −
16 𝔄_{10}  2  +
16 2.𝔄_{10}  ≠ 2,5  +
16 𝔄_{11}  2 b11  ∘
16 2.𝔄_{11}  11  +
16 2.𝔄_{11} ≠ 2,11 b11  ∘
16 𝔄_{12} 2 z3 ∘
16 2.𝔄_{12}  3 i2,i5,r7,b11 ∘
16 4_{2}.L_{3}(4)  3 i1,r7  ∘
16 2.Sz(8)  13 y7  +
16 M_{11}  11  +
16 M_{11}  ≠ 3,11 b11  ∘
16 M_{12}  11  +
16 M_{12}  ≠ 3,11 b11  ∘
16 4.M_{22}  7 i1,r11  ∘
18 3.𝔄_{7}  5 z3,b7  ∘
18 S_{4}(4)  ≠ 2  +

d G  ℓ field ind
18 J_{3}  3 b5 +
18 3.J_{3}  5 z3 ∘
18 3.J_{3}  ≠ 3,5 z3,b5 ∘
19 𝔄_{8}  7  +
19 𝔄_{9}  7  +
19 L_{3}(4)  3,7  +
20 𝔄_{7}  2  −
20 2.𝔄_{7}  ≠ 2,3  −
20 𝔄_{8}  0,5  +
20 𝔄_{9}  2 i15 ∘
20 L_{3}(4) 0,5 +
20 4_{2}.L_{3}(4) ≠ 2,3 i1  ∘
20 U_{3}(5)  ≠ 5  −
20 U_{4}(2)  0,5  +
20 2.U_{4}(2)  0,5  −
20 2.U_{4}(2)  0,5 z3 ∘
20 U_{4}(3) 2 +
20 2.U_{4}(3) ≠ 2,3  −
20 4.U_{4}(3) ≠ 2,3 i1  ∘
20 M_{22}  11  +
20 J_{1}  2  +
20 HS  2  −
21 𝔄_{7}  0,7  +
21 3.𝔄_{7}  ≠ 2,3 z3 ∘
21 𝔄_{8}  0,7 i15 ∘
21 𝔄_{8} ≠ 2 +
21 𝔄_{9}  0,7 i15 ∘
21 𝔄_{9} 3,5 +
21 3.L_{3}(4) ≠ 2,3 z3 ∘
21 U_{3}(3)  0,7  +
21 U_{3}(3)  0,7 i1  ∘
21 U_{3}(5)  ≠ 2,5  +
21 3.U_{3}(5) ≠ 3,5 z3 ∘
21 U_{4}(3) ≠ 2,3 +
21 3_{1}.U_{4}(3) ≠ 2,3 z3 ∘
21 U_{6}(2) 3 +
21 3.U_{6}(2) ≠ 2,3 z3 ∘
21 S_{6}(2)  ≠ 2  +
21 M_{22}  ≠ 2,11  +
21 3.M_{22}  ≠ 2,3 z3 ∘
21 M_{23}  23  +
21 J_{2}  5  +
21 J_{2}  ≠ 2,5 b5 +
21 HS  5  +
21 M^{c}L  3,5  +
22 2.L_{3}(4)  3 b5 +
22 U_{6}(2)  ≠ 2,3  +
22 M_{23}  ≠ 2,23  +
22 M_{24}  3  +

d G  ℓ field ind
22 J_{1}  19 b5  +
22 HS  ≠ 2,5  +
22 M^{c}L  ≠ 3,5  +
22 Co_{3}  2  −
22 Co_{3}  3  +
22 Co_{2}  2  +
23 U_{4}(2)  5  +
23 M_{24}  ≠ 2,3  +
23 Co_{3}  ≠ 2,3  +
23 Co_{2}  ≠ 2  +
24 3.𝔄_{7}  0,2 z3,b7 ∘
24 6.𝔄_{7} 7 z3 ∘
24 6.𝔄_{7}  0,5 z3,b7 ∘
24 2.𝔄_{8} ≠ 2,7 b7 ∘
24 4_{1}.L_{3}(4)  3 i1,r7 ∘
24 12_{1}.L_{3}(4)  7 z12 ∘
24 12_{1}.L_{3}(4) 0,5 z12,b7 ∘
24 U_{4}(2)  0  +
24 S_{4}(7)  2 b7  ∘
24 2.S_{4}(7) ≠ 2,7 b7  ∘
24 2.Sz(8)  13 y7  +
24 M_{11}  3  +
24 12.M_{22}  11 z24,b7 ∘
24 Co_{1}  2  +
24 2.Co_{1}  ≠ 2  +
25 S_{4}(7)  ≠ 2,7 b7  ∘
25 ^{3}D_{4}(2)  3  +
26 𝔄_{9}  2  +
26 𝔄_{10}  2  +
26 2.L_{3}(4)  7  +
26 L_{4}(3)  ≠ 3  +
26 U_{3}(3)  7  +
26 S_{6}(2)  7  +
26 ^{3}D_{4}(2) ≠ 2,3  +
26 ^{2}F_{4}(2)′  2  +
26 ^{2}F_{4}(2)′  ≠ 2 i2  ∘
27 𝔄_{9}  ≠ 2,7  +
27 U_{3}(3)  0  +
27 S_{6}(2)  ≠ 2,7  +
27 3.O_{7}(3)  ≠ 3 z3  ∘
27 3.G_{2}(3)  ≠ 3 z3  ∘
27 ^{2}F_{4}(2)′  ≠ 2 i1  ∘
27 J_{1}  11 b5,c19 +
27 3.F i_{22}  2 z3,b11 ∘
28 𝔄_{8}  ≠ 2,5  +
28 𝔄_{9}  ≠ 2,3  +
28 𝔄_{10}  5  +
28 2.L_{3}(4)  5  +
28 2.L_{3}(4)  0,7 b5  +

d G  ℓ field  ind
28 4_{2}.L_{3}(4)  5 i1  ∘
28 4_{2}.L_{3}(4) ≠ 2,5 i1,b5  ∘
28 U_{3}(3)  0,7 i1  ∘
28 U_{3}(5)  ≠ 5  +
28 O_{8}^{+} (2)  ≠ 2  +
28 2.M_{22}  5 i1,r11  ∘
28 2.HS  5 i1,r11  ∘
28 Ru  2  +
28 2.Ru  ≠ 2 i1  ∘
29 L_{3}(5)  31  +
29 L_{5}(2)  31  +
29 M_{12}  11  +
30 L_{3}(5) ≠ 5,31  +
30 L_{5}(2) ≠ 2,31  +
30 U_{4}(2)  0,5  +
30 U_{4}(2)  0,5 z3  ∘
31 L_{3}(5)  ≠ 2,5  +
31 L_{3}(5)  ≠ 2,5 i1  ∘
31 J_{1}  7 c19  +
32 2.𝔄_{8}  5 z3,b7  ∘
32 2.𝔄_{12}  ≠ 2,3  −
32 𝔄_{13}  2 b13  +
32 2.𝔄_{13}  13  −
32 2.𝔄_{13} ≠ 2,13 b13  −
32 2.𝔄_{14}  7 r3,r6,r10,b5,b13,b33 −
32 U_{3}(3)  0,2 b7  ∘
32 2.U_{4}(2)  5 z3  ∘
32 2.M_{12}  ≠ 2,3  −
33 S_{4}(4)  5  +
33 O_{8}^{−} (2)  7  +
34 𝔄_{9}  5  +
34 𝔄_{10}  3,5  +
34 𝔄_{11}  3  +
34 U_{4}(3)  2  −
34 S_{4}(4)  ≠ 2,5  +
34 S_{6}(2)  3  +
34 O_{8}^{−} (2) ≠ 2,7  +
34 M_{12}  3  +
34 M_{22}  2  −
34 J_{1}  19 b5  +
35 𝔄_{7}  ≠ 2,3  +
35 𝔄_{8}  ≠ 2  +
35 𝔄_{9}  ≠ 2  +
35 𝔄_{10}  ≠ 2,3  +
35 𝔄_{10}  5 r21  +
35 L_{3}(4)  ≠ 2,3  +
35 U_{4}(3)  ≠ 2,3  +
35 S_{6}(2)  ≠ 2  +
35 S_{8}(2)  ≠ 2  +

d G  ℓ field ind
35 O_{8}^{+} (2)  ≠ 2  +
35 Sz(8)  13  +
35 Sz(8)  ≠ 2,13 c_{13} +
36 2.𝔄_{7} 0,3 −
36 6.𝔄_{7}  0 z3 ∘
36 𝔄_{10}  ≠ 2,5  +
36 𝔄_{11}  11  +
36 2.L_{3}(4)  ≠ 2,7  +
36 4_{2}.L_{3}(4)  ≠ 2 i1  ∘
36 6.L_{3}(4)  ≠ 2,3 z3 ∘
36 12_{2}.L_{3}(4) ≠ 2,3 z12 ∘
36 2.U_{4}(2)  0 z3 ∘
36 3_{2}.U_{4}(3)  ≠ 3 z3 ∘
36 12_{2}.U_{4}(3) ≠ 2,3 z12 ∘
36 6.M_{22}  11 z12 ∘
36 J_{2}  ≠ 5  +
36 2.J_{2}  3 i1  ∘
38 L_{4}(3)  2,5  +
39 L_{4}(3)  0,13  +
39 U_{3}(4)  5  +
39 U_{3}(4)  ≠ 2,5 b5 +
40 2.L_{4}(3)  ≠ 2,3  +
40 4_{1}.L_{3}(4)  3 i1  ∘
40 U_{4}(2)  0,5 z3 ∘
40 S_{4}(5)  ≠ 5  +
40 S_{4}(9)  2  −
40 2.S_{4}(9)  ≠ 2,3  −
40 2.S_{6}(2)  7  +
40 S_{8}(3)  2 z3 ∘
40 2.S_{8}(3)  ≠ 2,3 z3 ∘
40 2.Sz(8) 7 +
40 2.Sz(8)  ≠ 2,7 y7 +
41 𝔄_{9}  3  +
41 𝔄_{10}  3  +
41 S_{4}(9)  ≠ 2,3  +
41 S_{8}(3)  ≠ 2,3 z3 ∘
41 J_{2}  5  +
42 𝔄_{9}  0,7  +
42 𝔄_{10}  0,7  +
42 6.L_{3}(4)  0,7 z3,b5 ∘
42 U_{3}(7)  ≠ 7  −
42 U_{7}(2)  ≠ 2  −
43 𝔄_{8}  5  +
43 𝔄_{11}  5  +
43 𝔄_{12}  5  +
43 U_{3}(7)  ≠ 2,7  +
43 U_{3}(7)  ≠ 2,7 i1  ∘
43 U_{3}(7)  ≠ 2,7 z8 ∘
43 U_{5}(2)  5  +

d G  ℓ field ind
43 U_{7}(2)  ≠ 2,3 z3  ∘
43 J_{1}  19 b5  +
44 𝔄_{11}  ≠ 3,5  +
44 𝔄_{12}  2  +
44 4_{2}.L_{3}(4)  7 i1  ∘
44 U_{5}(2)  ≠ 2,5  +
44 M_{11}  ≠ 3,5  +
44 M_{12}  2  +
44 2.M_{12}  ≠ 2,5 i5  ∘
44 M_{23}  2 b7  ∘
44 M_{24}  2 b7  ∘
45 𝔄_{8}  7  +
45 𝔄_{8}  ≠ 2,7 b7  ∘
45 𝔄_{11}  ≠ 2,11  +
45 𝔄_{12}  3  +
45 L_{3}(4)  7  +
45 L_{3}(4)  ≠ 2,7 b7  ∘
45 3.L_{3}(4)  0,5 z3,b7 ∘
45 U_{4}(2)  0,5 z3  ∘
45 3_{2}.U_{4}(3)  7 z3  ∘
45 3_{2}.U_{4}(3) ≠ 3,7 z3,b7 ∘
45 M_{11}  ≠ 2,11  +
45 M_{12}  ≠ 2,11  +
45 M_{22}  7  +
45 M_{22}  ≠ 2,7 b7  ∘
45 3.M_{22}  7 z3  ∘
45 3.M_{22}  ≠ 3,7 z3,b7 ∘
45 M_{23}  7  +
45 M_{23}  ≠ 2,7 b7  ∘
45 M_{24}  7  +
45 M_{24}  ≠ 2,7 b7  ∘
45 J_{1}  7 c19 +
45 3.M^{c}L  5 z3,b7 ∘
45 3.ON  7  ∘
47 𝔄_{9}  7  +
48 2.𝔄_{8}  ≠ 2  −
48 𝔄_{9}  0,2  +
48 2.𝔄_{9}  3  +
48 2.𝔄_{9}  ≠ 2,3 i6  ∘
48 𝔄_{10}  2  +
48 2.𝔄_{10}  3  +
48 2.𝔄_{10}  ≠ 2,3 i6  ∘
48 12_{1}.L_{3}(4) 0,7 z12,b5 ∘
48 12_{2}.L_{3}(4) 0,7 z12,b5 ∘
48 12_{2}.L_{3}(4)  5 z12 ∘
48 3.U_{3}(5) ≠ 3,5 z3  ∘
48 2.S_{6}(2) ≠ 2,7  +
48 O_{8}^{+} (2)  3  +
48 2.Sz(8)  5 c_{13}  +

d G  ℓ field ind
48 12.M_{22}  5 z12,b11 ∘
49 S_{4}(4)  17  +
49 S_{6}(2)  3  +
49 M_{22}  3 b11  ∘
49 J_{1}  11 b5,c19 +
49 HS  3 i5,b11 ∘
50 S_{4}(4) ≠ 2,17  +
50 S_{8}(2) 3 +
50 O_{8}^{+} (2) ≠ 2,3  +
50 O_{8}^{−} (2)  3  +
50 2.J_{2}  3 b5  −
50 2.J_{2}  ≠ 2,3 i1  ∘
50 He  7  +
51 U_{4}(4)  5  +
51 U_{4}(4) ≠ 2,5 z5  ∘
51 S_{4}(4) ≠ 2,5 b5  +
51 S_{8}(2) ≠ 2,3  +
51 O_{8}^{−} (2) ≠ 2,3  +
51 He ≠ 7 b7 ∘
52 L_{4}(3) ≠ 2,3  +
52 U_{3}(4) ≠ 2,5 z5  ∘
52 U_{4}(4) ≠ 2,5  +
52 2.S_{4}(5) ≠ 2,5 b5  −
52 ^{3}D_{4}(2) ≠ 2 +
52 2.F_{4}(2) ≠ 2  +
53 𝔄_{12}  11  +
53 𝔄_{13}  11  +
53 M_{12}  11  +
54 𝔄_{12}  0,3,7  +
54 6.L_{3}(4)  7 z3  ∘
54 M_{12}  0,3  +
54 M_{22}  7  +
54 6.M_{22}  7 z3  ∘
55 𝔄_{10}  5  +
55 𝔄_{11}  5  +
55 𝔄_{12}  ≠ 2,3  +
55 𝔄_{13}  13  +
55 L_{3}(7)  3,19  +
55 U_{5}(2)  ≠ 2  +
55 U_{5}(2) ≠ 2,3 z3  ∘
55 M_{11}  ≠ 2,3  +
55 M_{12}  ≠ 2,3  +
55 M_{22}  ≠ 2,7  +
55 J_{1}  19 b5  +
55 HS  5  +
56 𝔄_{8}  0,7  +
56 2.𝔄_{8}  0,7 z3  ∘
56 2.𝔄_{8}  0,7 i15  ∘
56 𝔄_{9}  ≠ 2,3  +

d G  ℓ field ind
56 2.𝔄_{9}  ≠ 2,3  +
56 𝔄_{10}  5  +
56 2.𝔄_{10}  5 r6,r21 +
56 2.𝔄_{11}  5 r6,r21 +
56 4_{1}.L_{3}(4) ≠ 2,3 i1  ∘
56 L_{3}(7)  0,2  +
56 U_{3}(8)  ≠ 2  −
56 2.U_{4}(3) ≠ 2,3  +
56 2.U_{6}(2)  ≠ 2  +
56 S_{6}(2)  ≠ 2,3  +
56 2.S_{6}(2)  3 i5  ∘
56 2.O_{8}^{+} (2)  ≠ 2  +
56 2.Sz(8) ≠ 2,13 c_{13}  +
56 2.M_{22}  ≠ 2,5  +
56 4.M_{22}  ≠ 2 z8  ∘
56 J_{1}  2 b5  −
56 J_{1}  5  +
56 J_{1}  ≠ 5,19 b5  +
56 2.J_{2}  5  −
56 2.J_{2}  0,7 b5  −
56 HS  2  +
56 2.HS  ≠ 2,5  +
57 L_{3}(7)  ≠ 2,7  +
57 3.L_{3}(7) ≠ 3,7 z3  ∘
57 U_{3}(8) ≠ 2,3 z3 ∘
57 3.U_{3}(8) ≠ 2,3 z9  ∘
57 J_{2}  3 b5  +
58 U_{4}(2)  5  +
58 2.J_{2}  7 b5  −
60 4_{2}.L_{3}(4)  3 i1,r7 ∘
60 6.L_{3}(4)  0,5 z3,b7 ∘
60 12_{2}.L_{3}(4) 0,5 z12,b7 ∘
60 U_{4}(2)  0,5  +
60 2.U_{4}(2)  0,5  −
60 2.U_{4}(2)  0,5 z3  ∘
60 U_{5}(3)  ≠ 3  −
60 S_{4}(11)  2 b11 ∘
60 2.S_{4}(11) ≠ 2,11 b11 ∘
61 L_{6}(2)  3,7  +
61 U_{5}(3)  ≠ 2,3 i1  ∘
61 S_{4}(11) ≠ 2,11 b11 ∘
62 L_{6}(2)  0,5,31  +
62 S_{6}(5)  ≠ 5 b5  −
63 L_{3}(4)  5  +
63 L_{3}(4)  ≠ 2,5 b5  +
63 3.L_{3}(4)  5 z3  ∘
63 3.L_{3}(4)  0,7 z3,b5 ∘
63 U_{3}(4) 13 +
63 2.S_{6}(5) ≠ 2,5 b5  +

d G  ℓ field  ind
63 Sz(8)  5  +
63 J_{2}  ≠ 2,5  +
64 𝔄_{8}  0  +
64 2.𝔄_{8}  0  −
64 𝔄_{10}  2  +
64 2.𝔄_{10}  0,7  +
64 𝔄_{13}  2  −
64 𝔄_{13}  3  +
64 𝔄_{14}  2,3  +
64 2.𝔄_{14}  ≠ 2,7  −
64 𝔄_{15}  2 b15  ∘
64 2.𝔄_{15}  3,5  −
64 2.𝔄_{15}  ≠ 2,3,5 b15  ∘
64 𝔄_{16}  2 b7,b15,b39,b55 ∘
64 L_{3}(4)  0  +
64 2.L_{3}(4)  0,7  +
64 4_{1}.L_{3}(4)  0,7 i1  ∘
64 4_{2}.L_{3}(4)  0 i1  ∘
64 U_{3}(4)  0,3  +
64 U_{4}(2)  0  +
64 2.U_{4}(2)  0  −
64 S_{4}(5)  2  −
64 S_{4}(5)  3  +
64 2.S_{6}(2)  5  +
64 2.S_{6}(2)  0,7 i5  ∘
64 Sz(8)  0,7  +
64 2.Sz(8)  0,7  +
64 G_{2}(3)  ≠ 3 z3  ∘
64 G_{2}(4)  3  +
64 2.M_{22}  11  +
64 4.M_{22}  3 i1  ∘
64 J_{1}  11 b5,c19  +
64 J_{2}  2 b5  +
64 2.J_{2}  5  −
64 2.J_{2}  0 b5  −
64 Suz  3  +
65 𝔄_{13}  ≠ 2,3,11  +
65 L_{4}(3)  ≠ 2,3  +
65 U_{3}(4)  0,13 z5  ∘
65 U_{3}(4)  ≠ 2,3  +
65 S_{4}(5)  0,13  +
65 Sz(8)  ≠ 2,7 y7  +
65 G_{2}(4)  ≠ 2,3  +
66 𝔄_{10}  7  +
66 𝔄_{11}  7  +
66 𝔄_{13}  ≠ 2,13  +
66 𝔄_{14}  7  +
66 U_{5}(2)  ≠ 2,3 z3  ∘
66 M_{12}  ≠ 2,3  +

d G  ℓ field ind
66 6.M_{22} 7 z3 ∘
66 6.M_{22}  0,5,11 z3,b7 ∘
66 3.Suz ≠ 3 z3 ∘
69 J_{1}  11 b5,c19 +
70 𝔄_{8}  ≠ 2,3  +
70 2.L_{3}(4)  ≠ 2,3  +
70 U_{4}(3)  2 z3  ∘
70 2.U_{4}(3)  ≠ 2,3  +
70 2.U_{4}(3)  ≠ 2,3 z3  ∘
70 S_{6}(2)  ≠ 2,3  +
70 M_{22}  2 b11 ∘
70 J_{2}  5  +
70 J_{2}  0,7 b5  +
71 L_{3}(8)  73  +
72 2.𝔄_{9}  7 z3,r2 ∘
72 L_{3}(8)  ≠ 2,73  +
72 U_{3}(9)  ≠ 3  −
73 L_{3}(8)  ≠ 2,7 z7  ∘
73 U_{3}(9)  ≠ 2,3  +
73 U_{3}(9)  ≠ 3,5 z5  ∘
75 𝔄_{10}  0,5  +
75 U_{3}(4)  ≠ 2,13 d13 ∘
75 J_{1}  7  +
76 𝔄_{14}  13  +
76 𝔄_{15}  13  +
76 J_{1}  2  −
76 J_{1}  ≠ 7,11  +
77 𝔄_{14}  ≠ 2,3,13  +
77 ^{2}F_{4}(2)′  3  +
77 J_{1}  ≠ 2,3,19  +
77 J_{1}  ≠ 2,5 b5  +
77 HS  ≠ 2,5  +
77 F i_{22}  3  +
78 𝔄_{9}  2  +
78 𝔄_{14}  ≠ 2,7  +
78 𝔄_{15}  3,5  +
78 6_{1}.U_{4}(3)  5 z3  ∘
78 S_{4}(5)  ≠ 2,5 b5  +
78 S_{6}(3)  ≠ 3  +
78 O_{7}(3)  ≠ 3  +
78 G_{2}(3)  ≠ 3  +
78 G_{2}(4)  ≠ 2  +
78 ^{2}F_{4}(2)′  ≠ 2,3  +
78 M_{12}  5  +
78 3.M_{22}  5 z3  ∘
78 J_{3}  2 b5,b17 +
78 Suz  3  +
78 3.Suz  ≠ 2,3 z3  ∘
78 F i_{22}  2  +

d G  ℓ field ind
78 F i_{22} ≠ 2,3 +
80 4_{1}.L_{3}(4)  0,5 i1,r7 ∘
80 4_{2}.L_{3}(4)  0,5 i1,r7 ∘
80 2.U_{4}(2) 0,5 −
80 J_{3}  2 b17 +
81 U_{4}(2)  0  +
83 𝔄_{9}  5  +
83 L_{4}(4)  5,17  +
83 S_{6}(2)  5  +
83 O_{8}^{+} (2)  5  +
83 O_{8}^{−} (2)  17  +
84 𝔄_{9}  0,7  +
84 𝔄_{10}  ≠ 2,5  +
84 𝔄_{11}  11  +
84 3.L_{3}(4)  0,7 z3  ∘
84 12_{2}.L_{3}(4)  0,7 z12 ∘
84 L_{4}(4)  0,3,7  +
84 U_{3}(5)  ≠ 2,5  +
84 3.U_{3}(5) ≠ 3,5 z3  ∘
84 3_{1}.U_{4}(3)  2 z3  ∘
84 6_{1}.U_{4}(3)  0,7 z3  ∘
84 12_{1}.U_{4}(3) ≠ 2,3 z12 ∘
84 S_{4}(13)  2 b13 −
84 2.S_{4}(13) ≠ 2,13 b13 −
84 S_{6}(2)  0,7  +
84 O_{8}^{+} (2)  0,7  +
84 O_{8}^{−} (2)  0,5,7  +
84 2.M_{12}  3 i5,b11 ∘
84 3.M_{22} 11 z3 ∘
84 3.M_{22}  2 z3,b11 ∘
84 J_{2}  2  +
84 2.J_{2}  0,7  −
84 J_{3}  2,3 b19 ∘
85 L_{4}(4)  ≠ 2,3 z3  ∘
85 U_{8}(2)  3  +
85 U_{8}(2)  ≠ 2,3 z3  ∘
85 S_{4}(4)  ≠ 2,3  +
85 S_{4}(13) ≠ 2,13 b13 +
85 S_{8}(2)  ≠ 2,3  +
85 J_{2}  5  +
85 J_{3}  19  +
85 J_{3}  0,5,17 b19 ∘
86 U_{8}(2) ≠ 2,3 +
88 4.M_{22}  5 z8,b7 ∘
89 𝔄_{10}  7  +
89 𝔄_{11}  5  +
89 𝔄_{12}  5  +
89 𝔄_{15}  7  +

d G  ℓ field ind
89 𝔄_{16}  7  +
89 L_{3}(9)  7,13  +
89 L_{4}(3)  13  +
89 U_{4}(3)  7  +
89 S_{4}(5)  13  +
89 J_{1}  7 c19 +
89 J_{2}  7  +
90 𝔄_{10}  0,3  +
90 𝔄_{15}  ≠ 7,13  +
90 𝔄_{16}  2  +
90 2.L_{3}(4) ≠ 2,7  +
90 6.L_{3}(4)  0,5 z3 ∘
90 L_{3}(9)  0,2,5  +
90 L_{4}(3)  0,5  +
90 U_{4}(3)  0,5  +
90 3_{1}.U_{4}(3)  2 z3 ∘
90 6_{2}.U_{4}(3) ≠ 2,3 z3 ∘
90 S_{4}(5)  0,3  +
90 O_{7}(3)  2  +
90 G_{2}(3)  2  +
90 6.M_{22}  11 z12 ∘
90 J_{2}  ≠ 2,7  +
91 𝔄_{15}  ≠ 2,3,5  +
91 L_{3}(9)  ≠ 2,3  +
91 L_{3}(9)  ≠ 2,3 i1 ∘
91 L_{3}(9)  ≠ 2,3 z8 ∘
91 S_{6}(2) 3 +
91 S_{6}(3)  ≠ 2,3 z3 ∘
91 O_{7}(3)  ≠ 2,3  +
91 Sz(8)  ≠ 2,5  +
91 G_{2}(3)  ≠ 2,3  +
91 M_{12}  11  +
92 2.G_{2}(4)  5  −
94 L_{5}(2)  7  +
94 S_{6}(2)  7  +
96 L_{3}(5)  31  +
96 L_{3}(5)  ≠ 5,31 x31 ∘
96 L_{3}(7) 3 +
96 3.L_{3}(7) ≠ 3,7 z3 ∘
96 12.M_{22}  11 z12 ∘
98 S_{6}(2)  3  +
98 M_{12}  5  +
98 M_{22}  2  −
98 M_{22}  5  +
98 HS  5  +
99 M_{12}  ≠ 2,5  +
99 M_{22}  0,3,11  +
99 3.M_{22}  0,7,11 z3 ∘

d G  ℓ field  ind
100 𝔄_{11}  2  +
100 𝔄_{12}  2  +
100 U_{5}(2)  3  +
101 𝔄_{9}  7  +
101 𝔄_{10}  7  +
101 J_{2}  7  +
101 He  2 b7  ∘
103 𝔄_{16}  3,5  +
103 𝔄_{17}  3,5  +
103 G_{2}(3)  7  +
104 2.𝔄_{9}  3  +
104 𝔄_{16}  0,11,13  +
104 U_{3}(5)  2  −
104 U_{4}(5)  2  −
104 U_{4}(5)  3  +
104 U_{4}(5)  ≠ 3,5 z3  ∘
104 2.U_{4}(5) 0,7,13 z3  ∘
104 2.U_{4}(5) ≠ 2,5  −
104 S_{4}(5)  ≠ 5  +
104 2.S_{4}(5)  0,13  −
104 2.S_{6}(2)  3,5  +
104 O_{7}(3)  2  +
104 2.O_{8}^{+} (2)  3,5  +
104 2.Sz(8)  0,13  +
104 G_{2}(3)  0,13  +
104 2.G_{2}(4) ≠ 2,5 b5  −
104 M_{23}  3 b11,b23 ∘
104 M^{c}L  3 b11  ∘
104 He  5 r21  +
105 𝔄_{9}  ≠ 2,3  +
105 𝔄_{16}  ≠ 2  +
105 𝔄_{17}  17  +
105 U_{3}(5)  0,7  +
105 3.U_{3}(5)  0,7 z3  ∘
105 3_{1}.U_{4}(3) ≠ 2,3 z3  ∘
105 U_{4}(5)  0,7,13  +
105 S_{6}(2)  ≠ 2,3  +
105 S_{6}(3)  ≠ 2,3  +
105 O_{7}(3)  ≠ 2,3  +
105 3.M_{22}  0,5,7 z3,b11 ∘
106 J_{1}  11 b5,c19 +
108 2.M_{12}  11  +
109 𝔄_{11}  3  +
109 ^{2}F_{4}(2)′  5 r2,r3,b13 +
110 𝔄_{11}  0,5,11  +
110 U_{3}(11)  ≠ 11  −
110 U_{5}(2)  ≠ 2  −
110 U_{5}(2)  ≠ 2,3 z3  ∘
110 2.M_{12}  ≠ 2,3 i2  ∘

d G  ℓ field  ind
110 J_{3}  19 b17,y9  +
110 Suz  2 b5,b13,r21 +
111 U_{3}(11)  ≠ 2,11  +
111 U_{3}(11)  ≠ 2,11 i1  ∘
111 3.U_{3}(11) ≠ 3,11 z3  ∘
111 3.U_{3}(11) 0,5,37 z12  ∘
111 Ly  5  +
112 2.𝔄_{9}  0,7  +
112 2.S_{6}(2)  0,7  +
112 2.O_{8}^{+} (2)  0,7  +
112 J_{4}  2  +
114 6_{1}.U_{4}(3)  7 z3  ∘
115 𝔄_{9}  7  +
118 𝔄_{17}  2  +
118 𝔄_{18}  2  +
118 S_{8}(2)  3,5  +
119 𝔄_{17}  ≠ 2,3,5  +
119 L_{5}(3)  11  +
119 U_{5}(2)  11  +
119 S_{8}(2)  0,7,17  +
119 J_{1}  11 c19  +
120 𝔄_{9}  0,5  +
120 2.𝔄_{9}  0,5 z3  ∘
120 𝔄_{11}  ≠ 2,11  +
120 𝔄_{12}  3  +
120 𝔄_{17}  ≠ 2,17  +
120 𝔄_{18}  3  +
120 12_{1}.L_{3}(4)  0,5 z12  ∘
120 L_{5}(3)  ≠ 3,11  +
120 U_{4}(3)  2  +
120 2.U_{4}(3)  ≠ 2,3  +
120 4.U_{4}(3)  ≠ 2,3 i1  ∘
120 6_{1}.U_{4}(3)  0,5 z3  ∘
120 12_{1}.U_{4}(3) ≠ 2,3 z12  ∘
120 U_{5}(2)  0,5  +
120 2.U_{6}(2)  3  +
120 6.U_{6}(2)  ≠ 2,3 z3  ∘
120 S_{6}(2)  0,5  +
120 2.S_{6}(2)  ≠ 2,3  +
120 M_{12}  0,5  +
120 2.M_{12}  0,5  +
120 2.M_{22}  ≠ 2,11  +
120 6.M_{22}  0,5,11 z3  ∘
120 12.M_{22}  ≠ 2,3 z24  ∘
120 M_{23}  2  +
120 M_{24}  2  +
120 J_{1}  ≠ 11,19 c19  +
120 2.HS  5 i1  ∘
121 L_{5}(3)  ≠ 2,3  +

d G  ℓ field  ind
121 S_{10}(3)  ≠ 3 z3  ∘
122 2.S_{10}(3) ≠ 2,3 z3  ∘
123 L_{5}(2)  5  +
124 𝔄_{10}  7  +
124 L_{3}(5)  ≠ 5  +
124 L_{3}(5)  2  −
124 L_{3}(5)  ≠ 2,5 i1  ∘
124 L_{3}(5)  0,31 y24′  ∘
124 L_{5}(2) 0,3,31  +
124 U_{3}(5)  7  +
124 Sz(32)  ≠ 2 i1  ∘
124 G_{2}(5)  ≠ 5  +
124 ^{2}F_{4}(2)′  3 b13  +
124 J_{2}  7  +
125 L_{3}(5)  0  +
125 L_{7}(2)  127  +
125 U_{3}(5)  0  +
126 𝔄_{10}  ≠ 2,5  +
126 𝔄_{11}  11  +
126 𝔄_{11}  ≠ 2,11 b11  ∘
126 𝔄_{12}  3 b11,i35  ∘
126 L_{7}(2) ≠ 2,127  +
126 U_{3}(5)  ≠ 2,5  +
126 U_{3}(5)  ≠ 2,5 i2  ∘
126 3.U_{3}(5)  0,7 z3  ∘
126 3.U_{3}(5)  0,7 z3,i2  ∘
126 3_{2}.U_{4}(3) ≠ 2,3 z3  ∘
126 6_{1}.U_{4}(3) ≠ 2,3 z3  ∘
126 6_{2}.U_{4}(3) ≠ 2,3 z3  ∘
126 6_{2}.U_{4}(3) ≠ 2,3 z12  ∘
126 S_{4}(7)  2  +
126 S_{4}(7)  ≠ 2,7  +
126 2.M_{22}  11  +
126 2.M_{22} ≠ 2,11 b11  ∘
126 6.M_{22}  0,5,7 z3,b11  ∘
126 J_{2}  0,7  +
126 2.J_{2}  ≠ 2,5 b5  −
126 3.J_{3}  2 z3,b17,b19  ∘
126 3.M^{c}L  11 z3  ∘
126 3.M^{c}L ≠ 3,11 z3,b11  ∘
126 Co_{3}  3 i5,b11,b23  ∘
128 2.𝔄_{11}  11  +
128 2.𝔄_{12}  11  +
128 2.𝔄_{16}  ≠ 2  +
128 𝔄_{17}  2 b17  +
128 2.𝔄_{17}  17  +
128 2.𝔄_{17} ≠ 2,17 b17  +
128 2.𝔄_{18}  3 r2,r5,r14,b17,b65,b77 +
130 S_{4}(5)  0,13  +

d G  ℓ field ind
131 𝔄_{11}  3,7  +
131 𝔄_{12}  3,7  +
131 L_{3}(11)  7,19  +
132 𝔄_{11}  0,11  +
132 𝔄_{12}  0,11  +
132 L_{3}(11) 0,2,3,5  +
132 12_{1}.U_{4}(3)  5 z12  ∘
132 HS  2  −
132 HN  2 b5  +
133 𝔄_{9}  5  +
133 𝔄_{10}  5 r21  +
133 𝔄_{11}  5 r21  +
133 L_{3}(11) ≠ 2,11  +
133 L_{3}(11) ≠ 5,11 z5  ∘
133 U_{3}(8)  ≠ 2  +
133 S_{6}(2)  5  +
133 M_{22}  5  +
133 J_{1}  ≠ 2,11  +
133 J_{1}  0,7,19 b5  +
133 J_{2}  3  +
133 HS  5  +
133 Ru  5  +
133 HN  5  +
133 HN  ≠ 2,5 b5  +
134 𝔄_{9}  5  +
134 𝔄_{18}  17  +
134 𝔄_{19}  17  +
134 S_{8}(2)  17  +
135 𝔄_{18}  ≠ 2,17  +
135 S_{8}(2) ≠ 2,17  +
136 𝔄_{18}  ≠ 2,3  +
136 𝔄_{19}  19  +
140 U_{4}(3)  ≠ 2,3  +
140 4.U_{4}(3) ≠ 2,3 i1  ∘
141 S_{6}(2)  5  +
142 Suz  2  +
143 𝔄_{12}  3  +
143 𝔄_{13}  3  +
143 Suz  ≠ 2,3  +
144 𝔄_{11}  2  +
144 2.𝔄_{11}  0,3,7  +
144 𝔄_{12}  5  +
144 𝔄_{12}  2 i35,z3 ∘
144 2.𝔄_{12}  3 i2,i5,r7 ∘
144 𝔄_{13} 5 +
144 𝔄_{13}  2 i35,z3 ∘
144 2.𝔄_{13}  3 i2,i5,r7 ∘
144 U_{3}(5) ≠ 5,7 b7 ∘
144 3.U_{3}(5)  0,2 z3,b7 ∘

d G  ℓ field ind
144 S_{4}(17)  2 b17 −
144 2.S_{4}(17) ≠ 2,17 b17 −
144 M_{12}  0,2  +
144 4.M_{22}  7 i1  ∘
144 4.M_{22}  0,3,11 i1,r7 ∘
144 12.M_{22}  7 z12 ∘
144 12.M_{22} 0,5,11 z12,b7 ∘
145 S_{4}(17) ≠ 2,17 b17 +
147 O_{8}^{+} (2) 3 +
150 3_{1}.U_{4}(3)  2 z3,b7 ∘
150 S_{4}(7)  ≠ 2,7 b7  ∘
151 𝔄_{19}  3  +
151 𝔄_{20}  3  +
152 2.𝔄_{10}  7  +
152 𝔄_{19}  ≠ 3,17  +
152 𝔄_{20}  2  +
152 L_{3}(7)  ≠ 3,7  +
152 2.O_{8}^{+} (2)  7  +
153 𝔄_{12}  5  +
153 𝔄_{19}  ≠ 2,19  +
153 𝔄_{20}  5  +
153 3_{2}.U_{4}(3)  5 z3  ∘
153 S_{4}(4)  ≠ 2  +
153 O_{10}^{−}(2)  3,5  +
153 3.M_{22}  5 z3  ∘
153 J_{3}  3 b5  +
153 3.J_{3}  5 z3  ∘
153 3.J_{3}  ≠ 3,5 z3,b5 ∘
153 3.M^{c}L  5 z3,b7 ∘
153 He  7  +
153 He  ≠ 2,7 b7  ∘
153 3.ON  2 z3  ∘
154 𝔄_{12}  0,7,11  +
154 L_{4}(5)  2,3,13  +
154 O_{8}^{−} (2)  3  +
154 O_{10}^{−}(2) ≠ 2,3,5  +
154 M_{22}  0,7,11  +
154 2.M_{22}  ≠ 2,5 i1  ∘
154 HS  ≠ 2,5  +
154 ON  3 r7  +
155 𝔄_{10}  5  +
155 𝔄_{11}  7  +
155 L_{3}(5)  0,31  +
155 L_{3}(5)  0,31 i1  ∘
155 L_{4}(5)  0,31  +
155 L_{5}(2)  ≠ 2  +
155 S_{10}(2)  ≠ 2  +
155 O_{10}^{+}(2)  ≠ 2  +
156 2.L_{4}(5) ≠ 2,5  +

d G  ℓ field ind
156 4.L_{4}(5)  ≠ 2,5 i1  ∘
156 U_{3}(13)  ≠ 13  −
156 S_{4}(5)  ≠ 2,5  +
156 2.S_{4}(5)  ≠ 2,5  −
157 U_{3}(13) ≠ 2,13  +
157 U_{3}(13) ≠ 7,13 z7  ∘
160 𝔄_{9}  2  +
160 2.𝔄_{9}  0,5  +
160 𝔄_{10}  ≠ 3,7  +
160 2.𝔄_{10}  5 r6,r21 +
160 2.𝔄_{12}  0,5,7 b11 ∘
160 2.O_{8}^{+} (2) 0,5 +
160 2.M_{12}  0,5 b11 ∘
160 4.M_{22} 11 i1 ∘
160 4.M_{22}  ≠ 2,11 i1,r11 ∘
160 J_{2}  0,2  +
162 𝔄_{9}  0,3  +
162 3.G_{2}(3)  2 z3,b13 ∘
164 𝔄_{11}  2  −
164 𝔄_{12}  2  −
165 𝔄_{11}  0,11  +
165 𝔄_{12}  ≠ 2,3  +
165 𝔄_{13}  13  +
165 U_{5}(2)  ≠ 2,3  +
167 S_{6}(3)  13  +
167 O_{7}(3)  13  +
167 G_{2}(3)  13  +
168 𝔄_{9}  0,7  +
168 2.𝔄_{9}  5  +
168 2.𝔄_{9}  0,7 i15  ∘
168 2.𝔄_{10}  5 r21 +
168 S_{6}(2)  ≠ 2,3  +
168 2.S_{6}(2)  ≠ 2,3  +
168 S_{6}(3)  ≠ 3,13  +
168 O_{7}(3)  0,5,7  +
168 2.O_{8}^{+} (2)  5  +
168 G_{2}(3)  0,7  +
169 𝔄_{20}  19  +
169 𝔄_{21}  19  +
170 𝔄_{20}  ≠ 2,3,19  +
170 U_{9}(2)  ≠ 2  −
171 𝔄_{20}  ≠ 2,5,  +
171 𝔄_{21}  3,7  +
171 3.U_{9}(2)  ≠ 2,3 z3  ∘
171 S_{6}(7)  ≠ 7 b7  ∘
171 O_{8}^{−} (2)  7  +
171 3.J_{3}  ≠ 2,3 z3  ∘
171 3.J_{3}  0,17,19 z3,b5 ∘
172 2.S_{6}(7)  ≠ 2,7 b7  ∘

d G  ℓ field ind
174 S_{4}(7) 2 +
174 6.M_{22}  11 z12 ∘
174 HS  11  +
175 S_{4}(7)  ≠ 2,7  +
175 O_{8}^{+} (2)  ≠ 2,3  +
175 J_{2}  ≠ 2,3  +
175 HS  0,5,7  +
176 U_{5}(2)  ≠ 2,3  +
176 2.U_{6}(2)  ≠ 2,3  +
176 M_{12}  0,11  +
176 4.M_{22}  0,11 i1 ∘
176 2.HS  ≠ 2,5 i1 ∘
176 2.F i_{22}  3 b13 +
180 S_{4}(19)  2 b19 ∘
180 2.S_{4}(19) ≠ 2,19 b19 ∘
181 L_{3}(13) 3,61 +
181 S_{4}(19)  ≠ 2,19 b19 ∘
182 L_{3}(13)  0,2,7  +
182 U_{6}(3)  ≠ 3  −
182 2.U_{6}(3)  ≠ 2,3 i1 ∘
182 2.S_{6}(3) ≠ 2,3 −
182 2.S_{6}(3)  ≠ 2,3 z3 ∘
182 O_{7}(3)  ≠ 2,3  +
182 G_{2}(3)  ≠ 2,3  +
183 L_{3}(13)  ≠ 2,13  +
183 L_{3}(13)  ≠ 2,13 i1 ∘
183 3.L_{3}(13) 0,7,61 z12 ∘
183 3.L_{3}(13) ≠ 3,13 z3 ∘
183 U_{6}(3)  ≠ 2,3  +
185 O_{10}^{+}(2)  3,17  +
186 𝔄_{11}  2  +
186 L_{3}(5)  ≠ 2,5  +
186 S_{10}(2)  3  +
186 O_{10}^{+}(2) ≠ 2,3,17  +
186 O_{10}^{−}(2)  3  +
187 S_{10}(2)  ≠ 2,3  +
187 O_{10}^{−}(2)  ≠ 2,3  +
188 𝔄_{11}  5  +
188 𝔄_{21}  2,5  +
188 𝔄_{22}  2,5  +
188 U_{4}(3)  5  +
189 𝔄_{9}  ≠ 2,5  +
189 𝔄_{21}  ≠ 2,5,19  +
189 L_{4}(4)  5  +
189 L_{4}(4)  ≠ 2,5 b5 +
189 3.U_{3}(8)  ≠ 2,3 z3 ∘
189 U_{4}(3) 0,7 +
189 3_{2}.U_{4}(3) ≠ 3,5 z3 ∘
189 S_{6}(2)  ≠ 2,5  +

d G  ℓ field ind
189 3.G_{2}(3) 13 z3 ∘
189 3.G_{2}(3)  0,7 z3,b13 ∘
189 J_{2}  5  +
189 J_{2}  ≠ 2,5 b5  +
190 𝔄_{21}  ≠ 2,3,7  +
190 𝔄_{22}  11  +
190 M_{22}  11  +
190 2.J_{2}  5  −
194 S_{6}(3)  7  +
194 O_{7}(3)  7  +
195 S_{6}(3)  0,5,13  +
195 O_{7}(3)  0,5,13  +
196 𝔄_{13}  5  +
196 𝔄_{14}  5  +
196 S_{4}(8)  ≠ 2  +
196 S_{6}(2)  3  +
196 ^{3}D_{4}(2)  ≠ 2  +
198 𝔄_{10}  2  +
198 𝔄_{11}  2  +
199 𝔄_{10}  7  +
199 𝔄_{11}  7  +
199 J_{2}  7  +
200 𝔄_{10}  2  +
200 2.S_{4}(7)  ≠ 2,7 b7  ∘
201 S_{6}(2)  7  +
202 2.J_{2}  5  −
203 S_{8}(2)  3  +
203 O_{8}^{−} (2)  3,5  +
204 3_{1}.U_{4}(3)  2 z3  ∘
204 U_{5}(4)  ≠ 2  ∘
204 S_{4}(4)  ≠ 2,5 b5  +
204 O_{8}^{−} (2)  ≠ 2,3  +
205 5.U_{5}(4)  ≠ 5 z5  ∘
207 𝔄_{13}  11  +
207 S_{4}(4)  17  +
208 𝔄_{13}  ≠ 3,5,11  +
208 𝔄_{14}  2  +
208 𝔄_{22}  3,7  +
208 𝔄_{23}  3,7  +
208 L_{4}(3)  2,5  +
208 2.L_{4}(3)  ≠ 2,3 i2  ∘
208 S_{4}(5)  ≠ 5 b5  +
208 2.S_{4}(5)  ≠ 2,5 b5  −
208 M_{23}  7  +
208 2.Suz  3  −
209 𝔄_{22}  ≠ 2,3,5,7  +
209 J_{1}  0,11,19  +
210 𝔄_{10}  0,7  +
210 𝔄_{11}  ≠ 2,11  +

d G  ℓ field ind
210 𝔄_{12}  3  +
210 𝔄_{22}  ≠ 2,11  +
210 𝔄_{23}  23  +
210 U_{4}(3)  ≠ 2,3  +
210 2.U_{4}(3)  ≠ 2,3 i1  ∘
210 3_{1}.U_{4}(3)  ≠ 2,3 z3  ∘
210 6_{1}.U_{4}(3)  ≠ 2,3 z3  ∘
210 U_{6}(2)  3  +
210 3.U_{6}(2)  ≠ 2,3 z3  ∘
210 S_{6}(2)  ≠ 2,3  +
210 O_{8}^{+} (2)  ≠ 2,3  +
210 M_{22}  ≠ 2,11  +
210 2.M_{22}  ≠ 2,11  +
210 3.M_{22}  ≠ 2,3 z3  ∘
210 6.M_{22}  0,5,7 z3  ∘
210 6.M_{22}  ≠ 2,3 z12 ∘
210 M_{23}  23  +
210 HS  5  +
210 M^{c}L  3,5  +
214 J_{3}  19 b17,y9 +
216 𝔄_{9}  0  +
216 2.𝔄_{10}  0,3  +
216 12_{1}.U_{4}(3)  0,7 z12 ∘
216 12_{2}.U_{4}(3) ≠ 2,3 z12 ∘
216 S_{6}(2)  0  +
216 2.J_{2}  0,3  −
217 𝔄_{10}  5  +
217 L_{5}(2)  ≠ 2  +
217 L_{6}(2)  ≠ 2  +
218 ^{3}D_{4}(3)  2  +
218 ^{3}D_{4}(3)  73  +
219 ^{3}D_{4}(3) ≠ 2,3,73  +
220 𝔄_{13}  ≠ 2,13  +
220 𝔄_{14}  7  +
220 U_{4}(4)  5  +
220 U_{5}(2)  ≠ 2,3 z3  ∘
220 M_{23}  2 b7,b23 ∘
220 M_{24}  2 b7,b23 ∘
220 2.Suz  ≠ 2,3  −
221 𝔄_{12}  7  +
221 𝔄_{13}  7  +
221 U_{4}(4)  ≠ 2 b5  +
223 S_{4}(7)  5  +
224 2.𝔄_{9}  0,7  +
224 𝔄_{10}  ≠ 2,5  +
224 4.U_{4}(3)  ≠ 2,3 z12 ∘
224 S_{4}(7)  ≠ 2,5,7  +
224 2.O_{8}^{+} (2)  ≠ 2,5  +
224 J_{2}  0,7 b5  +

d G  ℓ field ind
225 𝔄_{10} 0,5 +
225 S_{4}(4)  ≠ 2,17 d17 +
225 J_{2}  ≠ 2,7  +
229 𝔄_{23}  11  +
229 𝔄_{24}  11  +
229 U_{6}(2)  3  +
229 M_{23}  11  +
229 M_{24}  11  +
230 𝔄_{23}  ≠ 3,7,11  +
230 𝔄_{24}  2  +
230 M_{23}  0,5,23  +
230 M^{c}L  2,5  +
230 Co_{3}  2,5  +
230 Co_{2}  2  +
231 𝔄_{11}  0,7,11  +
231 𝔄_{23}  ≠ 2,23  +
231 𝔄_{24}  3  +
231 U_{6}(2)  ≠ 2,3  +
231 3.U_{6}(2) ≠ 2,3 z3 ∘
231 M_{22} ≠ 2,5 +
231 3.M_{22}  ≠ 2,3 z3 ∘
231 M_{23} ≠ 2,23 +
231 M_{23}  ≠ 2,3,5 i15 ∘
231 M_{24} 3,5 +
231 M_{24}  ≠ 2,3,5 i15 ∘
231 HS  ≠ 2,5  +
231 M^{c}L  0,7,11  +
231 Co_{3}  3  +
233 𝔄_{13}  5  +
233 𝔄_{14}  5  +
234 L_{4}(3)  0,13  +
236 2.J_{2}  3  −
238 S_{8}(2)  ≠ 2,3  +
240 U_{3}(16)  ≠ 2  −
241 U_{3}(16) ≠ 2,17 z17 ∘
244 J_{3}  2 b17 +
245 O_{8}^{−} (3) 13 +
246 O_{8}^{−} (3) ≠ 3,13  +
246 ^{2}F_{4}(2)′ 2 +
246 He  2 b17 +
248 S_{4}(5)  2 b5 +
248 T h  all  +

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Low-dimensional representations

61. E. Wings, ‘Die Zerlegungszahlen der Hauptblöcke der unitären Gruppen GU_{4}(q^{2}) in nicht-definierender Charakteristik.’ Diplomarbeit, Lehrstuhl D für Mathematik, RWTH Aachen, (1990).

Gerhard Hiss Gerhard.Hiss@Math.RWTH-Aachen.DE

Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany

Gunter Malle malle@mathematik.uni-kassel.de

FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany