ISSN 2148-838X J. Algebra Comb. Discrete Appl. 9(3) • 149–159 Received: 7 November 2020 Accepted: 21 March 2022 Journal of Algebra Combinatorics Discrete Structures and Applications The covering numbers of the McLaughlin group and some primitive groups of low degree Research Article Michael Epstein Abstract: A finite cover of a group G is a finite collection C of proper subgroups of G with the property that⋃ C = G. A finite group admits a finite cover if and only if it is noncyclic. More generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. If C is a finite cover of a group G, and no cover of G with fewer subgroups exists, then C is said to be a minimal cover of G, and the cardinality of C is called the covering number of G, denoted by σ(G). Here we investigate the covering numbers of the McLaughlin sporadic simple group and some low degree primitive groups. 2010 MSC: 20D60 Keywords: Covering numbers, McLaughlin group, Primitive groups 1. Introduction A finite collection C of proper subgroups of a group G with the property that ⋃ C = G is called a finite cover of G. If no cover of G with fewer subgroups exists, then C is called a minimal cover and the number of subgroups in C is called the covering number of G, denoted σ(G). A number of interesting results about covering numbers were proven in [6], where it was conjectured that the covering number of a finite, noncyclic, solvable group is of the form 1 + q, where q is the order of a chief factor of the group. This conjecture was proven by M. J. Tomkinson in [18]. In light of this result, much of the more recent work on covering numbers of finite groups has focused on nonsolvable groups, and in particular on simple and almost simple groups. A number of results can be found in [2, 3, 5, 10, 13–17]. In this article, we investigate group covering numbers in some outstanding open cases from [11] and [13]. In Section 31, we determine the covering number of the McLaughlin sporadic simple group. This was first attempted by P. E. Holmes in [13], where it was shown that 24541 ≤ σ(McL) ≤ 24553. We will show that the upper bound of 24553 is in fact the correct covering number. Michael Epstein; Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA (email: michael.epstein@colostate.edu). 1 Section 3 of this article is based on a portion of the author’s dissertation [9], submitted in partial fulfillment of the requirements for the degree of doctor of philosophy at Florida Atlantic University. 149 https://orcid.org/0000-0002-5541-2323 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 In [11], the authors investigated integers which are not covering numbers of groups, and in the process determined the covering numbers of many primitive groups of degree less than or equal to 129. However, a few difficult cases remain unsolved (see Table 2 of [11] for upper and lower bounds). In Section 4 we consider some of these open cases. In particular, we find the exact covering numbers of L5(3), PΣL2(121), A5 wr 3, (A5 ×A5 ×A5).6, PΓL2(125), and L7(2) and give improved bounds for the covering numbers of A7 wr 2, L3(4).222, HS : 2, L2(11) wr 2, and PGU3(5). 2. Preliminaries In this article we follow [8] for basic group theoretic terminology and notation. We use the notation of [7] for simple groups and group structures. The basic method we use for determining the covering numbers of finite groups is based on two simple observations: First, that one need only consider covers consisting of maximal subgroups, and second, that in order to prove that a collection C of proper subgroups is a cover of a group G it is sufficient to show that each of the maximal cyclic subgroups is contained in a subgroup from C. We call the maximal cyclic subgroups of G the principal subgroups of G and a generator of a principal subgroup is called a principal element of G. In light of these observations, the first step in determining the covering number of a group G is to determine the maximal subgroups and principal subgroups of G up to conjugacy. The second step is to determine which principal subgroups are contained in which maximal subgroups. Consider a bipartite graph whose vertex set is the union of a conjugacy class P of principal subgroups of G and a conjugacy class M of maximal subgroups of G, with an edge between C ∈P and H ∈M if and only if C ≤ H. The group G acts transitively on both P and M by conjugation, so any two vertices in P have the same degree aP,M and any two vertices in M have the same degree bP,M. Moreover, counting the number of edges in the graph in two ways yields the equation aP,M|P| = bP,M|M|. (1) Consequently, one can construct two matrices A = (aP,M) and B = (bP,M), with rows indexed by the conjugacy classes of principal subgroups of G and columns indexed by the classes of maximal subgroups, describing the incidence between the principal subgroups of G and the maximal subgroups up to conju- gacy. In this article we give only the matrix A for each group as the entries tend to be smaller, and as noted above the entries of B can easily be computed from those of A. In order to compute the entries of the matrices A and B, we make use of the characters of the permutation representations of G acting on the sets of right cosets of representatives of each conjugacy class of maximal subgroups. This is justified by the following well-known fact from character theory: Proposition 2.1. Let H be a subgroup of G, x ∈ G, K be the conjugacy class of x, and θ be the permutation character of the action of G on the right cosets of H. Then, |K ∩H| = θ(x)|K| |G : H| . With this proposition we can compute the entries of A and B as follows: Proposition 2.2. Let x ∈ G, P be the conjugacy class of 〈x〉, H be a subgroup of G from conjugacy class M, and θ be the permutation character of the action of G on the right cosets of H. Then aP,M = θ(x)|M| |G : H| and bP,M = θ(x)|P| |G : H| . 2 For clarity we use the same numbering for the groups L3(4).2 as in [11]. In [7], this group is called L3(4).21. 150 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 Proof. Let K be the conjugacy class of x. We consider another bipartite graph, with vertex set K∪M and with an edge between y ∈ K and J ∈ M if and only if y ∈ J. Any two vertices in K have the same degree α, any two vertices in M have the same degree β, and α|K| = β|M|. By Proposition 2.1, β = θ(x)|K| |G:H| , and since y ∈ J if and only if 〈y〉 ≤ J, aP,M = α = β|M| |K| = θ(x)|M| |G:H| . The formula for bP,M then follows from (1). Sometimes it is possible to simplify the problem and reduce the number of conjugacy classes of subgroups we must consider. For example, if the members of some conjugacy class of maximal subgroups contain no principal subgroups, then this conjugacy class may be eliminated from further consideration as no minimal cover can contain subgroups from this class. On the other hand, define U to be the set of all maximal subgroups H of G for which there is a principal subgroup C such that H is the unique maximal subgroup of G containing C. Then U is a (possibly empty) union of conjugacy classes of maximal subgroups of G and any cover of G consisting of maximal subgroups must contain every member of U. As we only consider covers consisting of maximal subgroups, we must include every member of U. Consequently, we need no longer consider the conjugacy classes of subgroups in U, nor the conjugacy classes of principal subgroups covered by the members of U. Thus we may delete the rows and columns of A and B corresponding to the classes of subgroups that have been removed from consideration to obtain submatrices A′ and B′, of A and B respectively, which describe the incidence between the remaining classes of subgroups. The third step is to find a small cover which will be a candidate for a minimal cover. This will often, but not always, be a union of conjugacy classes of maximal subgroups of G. One can easily find covers of this type; they consist of the subgroups from U as well as those from the conjugacy classes of maximal subgroups corresponding to a set of columns of A′ whose sum has no zero entries. In any case, once a cover has been found, the number of subgroups in the cover gives an upper bound on σ(G). We establish a lower bound for the covering number by solving a certain integer linear programming (ILP) problem. To make this more explicit, consider a cover C consisting of maximal subgroups of G, and let M1, . . . ,Mn be the remaining conjugacy classes of maximal subgroups of G (i.e. those not contained in U). Define xj = |Mj ∩C| for 1 ≤ j ≤ n. For each remaining conjugacy class P of principal subgroups of G the following inequality must hold: bP,M1x1 + bP,M2x2 + · · · + bP,Mnxn ≥ |P|. The sum of |U| and the minimum value of the function x1 +· · ·+xn subject to these linear constraints, as well as the conditions xj ∈ Z, and 0 ≤ xj ≤ |Mj| for 1 ≤ j ≤ n, is a lower bound for σ(G). If this lower bound is equal to the number of subgroups in a cover C, then necessarily C is minimal and σ(G) = |C|, though in general this bound may be strictly less than the actual covering number of the group. One may be able to work around this difficulty in some instances by deriving additional linear constraints by, for example, considering how the group acts on appropriate combinatorial objects. This is illustrated in the computation of the covering number of the McLaughlin group in Section 3. In cases where the lower bound described above is less than the number of subgroups in the best cover we have found, we can formulate a different linear programming problem to find the covering number. Let M be the matrix with rows and columns indexed by the remaining principal subgroups and the remaining maximal subgroups of G respectively, such that the entry in the row C and column H is 1 if C ≤ H and 0 otherwise. Then a minimal cover of G consisting of maximal subgroups is then of the form U ∪X, where the members of X are the maximal subgroups corresponding to the positions of the 1’s in a (0,1)-vector x whose entries have minimum possible sum subject to the condition that Mx ≥~1, where ~1 denotes the column vector of all 1’s of length equal to the number of rows of M. The benefit of this second ILP formulation is that a solution to the linear programming problem always yields the exact covering number of the group, but the trade off is that it generally has far more variables and constraints than the other linear programming problem described above, and is usually much more difficult to solve. However, even if the problem is intractable, we can get upper and lower bounds on the covering number from an incomplete attempt to solve the ILP from the best incumbent solution and best bound found 151 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 by the solver at the time it is interrupted. We take this approach in Section 4 when investigating the covering numbers of low degree primitive groups. 3. The McLaughlin group The McLaughlin group, one of the sporadic simple groups, is a subgroup of index two in the full automorphism group of the McLaughlin graph, a strongly regular graph with parameters (275, 112, 30, 56). We note that the independence number of the McLaughlin graph is 22 (see [4]), and that the McLaughlin group acts transitively on the vertices, the edges, and the nonedges of the McLaughlin graph. The maximal subgroups from classes M1, M6, and M7 are the stabilizers of the vertices, edges, and nonedges of the McLaughlin graph respectively. Tables 1 and 2 give the conjugacy classes of principal and maximal subgroups of the McLaughlin group, and the matrix A for the McLaughlin group is given in Table 3. Table 1. Conjugacy classes of principal subgroups of the McLaughlin group Class Order Class Size P1 5 8981280 P2 6 12474000 P3 8 28066500 P4 9 11088000 P5 11 16329600 P6 12 18711000 P7 14 21384000 P8 30 7484400 Table 2. Conjugacy classes of maximal subgroups of the McLaughlin group Class Order Class Size Structure M1 3265920 275 U4(3) M2 443520 2025 M22 M3 443520 2025 M22 M4 126000 7128 U3(5) M5 58320 15400 31+4+ : 2S5 M6 58320 15400 34 : M10 M7 40320 22275 L3(4) : 2 M8 40320 22275 2.A8 M9 40320 22275 24 : A7 M10 40320 22275 24 : A7 M11 7920 113400 M11 M12 3000 299376 51+2+ : 3 : 8 152 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 Table 3. Matrix A for McL M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 P1 5 5 5 3 0 5 5 0 5 5 5 1 P2 2 3 3 3 2 1 6 7 6 6 6 0 P3 1 1 1 2 2 2 1 1 1 1 2 4 P4 2 0 0 0 1 1 0 0 0 0 0 0 P5 0 1 1 0 0 0 0 0 0 0 1 0 P6 1 0 0 0 3 2 0 1 0 0 0 2 P7 0 0 0 0 0 0 1 1 1 1 0 0 P8 0 0 0 0 1 0 0 1 0 0 0 1 Theorem 3.1. The covering number of the McLaughlin group is 24553. Proof. We show that there exists a cover C ⊆ M1 ∪M2 ∪M8 with |C| = 24553 as follows: First, observe that M2∪M8 is sufficient to cover all of the principal elements of McL except for those of order 9, which generate the cyclic subgroups from P4, and that these are contained in the subgroups from class M1. Each element of order 9 fixes two adjacent vertices in the McLaughlin graph, the edge between them, and no nonedges. As the independence number of the McLaughlin graph is 22, we may choose an independent set I consisting of 22 vertices of the McLaughlin graph. Let S be the subset of M1 consisting of the stabilizers of the vertices of the McLaughlin graph which are not in I. Then |S| = 253, and given any element x of order 9 in McL, at least one of the two subgroups from M1 which contain x is in S. Consequently, C = S ∪M2 ∪M8 is a cover of McL with |C| = 253 + 2025 + 22275 = 24553. As a result, σ(McL) ≤ 24553. We now prove that no smaller cover can exist. Suppose that C is a minimal cover of McL consisting of maximal subgroups. For 1 ≤ j ≤ 12, let xj = |C ∩Mj|. The cyclic subgroups from P5 of order 11 are contained only in the maximal subgroups from classes M2, M3, and M11. Each subgroup H ∈M2∪M3 contains 8064 of these cyclic subgroups and each subgroup H ∈M11 contains 144 of them. Since C covers all of the elements of order 11, 8064(x2 + x3) + 144x11 ≥ |P5| = 16329600, from which it follows that x2 + x3 + x11 ≥ 2025. Now, the cyclic subgroups of order 14 from P7 are contained only within the maximal subgroups from classes Mi for 7 ≤ i ≤ 10, and a subgroup from any of these classes contains exactly 960 members of P7. Hence, 960(x7 + x8 + x9 + x10) ≥ |P7| = 21384000, from which we may deduce that x7 + x8 + x9 + x10 ≥ 22275. A similar analysis for the subgroups from P4 will yield the inequality x1 + x5 + x6 ≥ 138, but this is insufficient for our purposes. We consider the action of the principal subgroups of McL of order 9 on the McLaughlin graph to obtain a stronger inequality. Let W be the set of vertices of the McLaughlin graph whose stabilizers are not contained in C. Observe that x1 + |W | = 275. Now, an cyclic subgroup 〈x〉 of order 9 in McL is left uncovered by the subgroups from M1 ∩C if and only if the unique edge e of the McLaughlin graph fixed by x is an edge of the subgraph induced by W . The stabilizer in McL of an edge of the McLaughlin graph contains 720 cyclic subgroups of order 9, so the total number of members of P4 left uncovered by the subgroups from M1 ∩C is 720n, where n is the number of edges in the subgraph of the McLaughlin graph induced by W . Aside from the subgroups from class M1, the only maximal subgroups which contain members of P4 are those from M5 ∪M6, each of which contains 720 of them. Consequently, at least n subgroups from M5 ∪M6 are needed to cover the remaining cyclic subgroups of order 9. Now, the McLaughlin graph has independence number 22, so the subgraph induced by W can have no independent set of more than 22 vertices. Therefore n ≥ |W |− 22. Thus, x1 + x5 + x6 ≥ x1 + n ≥ x1 + |W |− 22 = 275 − 22 = 253. 153 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 It follows from the inequalities x2 + x3 + x11 ≥ 2025, x7 + x8 + x9 + x10 ≥ 22275, and x1 + x5 + x6 ≥ 253 that σ(McL) = |C| = ∑12 j=1 xj ≥ 2025 + 22275 + 253 = 24553, which completes the proof. This settles one of two open cases for which upper and lower bounds were given in [13]. The other such case, that of the Janko group J1, remains open, though improved bounds were given in [16]. 4. Some primitive groups of low degree In this section we take a computational approach to investigating the covering numbers of some of the remaining primitive groups of degree less than 129 from Table 2 of [11]. The approach is based on linear programming and follows the method described in Section 2. All of the group-theoretic computations were done using Magma [1], and the linear programming problems were solved using Gurobi optimization software [12]. In many cases we are unable to solve the larger ILP problem in a reasonable amount of time; in this case we can still obtain upper and lower bounds for the covering number from Gurobi’s best incumbent solution and best lower bound at the time the computation is interrupted. We summarize our results in Tables 4 and 5. Previous Previous Group Lower Bound Upper Bound Covering Number L5(3) 393030144 – 393031475 PΣL2(121) 671 794 794 A5 wr 3 216 342 317 (A5 ×A5 ×A5).6 1000 1217 1127 PΓL2(125) 7750 7876 7876 L7(2) 184308203520 – 184308218125 Table 4. Exact covering numbers of some of the low degree primitive groups from the list of open cases in [11]. Previous Previous New New Group Lower Bound Upper Bound Lower Bound Upper Bound A7 wr 2 447 667 460 667 L3(4).22 138 166 144 166 HS : 2 11859 22375 15127 22375 L2(11) wr 2 570 926 721 817 PGU3(5) 6000 6526 6307 6378 Table 5. Improved upper and lower bounds for the covering numbers of some of the low degree primitive groups from the list of open cases in [11]. We illustrate our method by presenting two cases in detail. The others are similar, and for these we simply describe a cover of size equal to the covering number given in Table 4 or the upper bound given in Table 5. 154 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 4.1. L5(3) L3(5) has 17 conjugacy classes of principal subgroups and 8 classes of maximal subgroups. These are as in Tables 6 and 7. We note that each member of P17 is contained in a unique maximal subgroup of L5(3), which is from M8, so we must use all 393030144 subgroups from that class in the cover. The matrix A′ is given in Table 8. Table 6. Conjugacy classes of principal subgroups of L5(3) Class Order Class Size P1 6 366949440 P2 6 366949440 P3 6 2201696640 P4 8 1857681540 P5 8 928840770 P6 9 489265920 P7 12 1238454360 P8 18 733898880 P9 24 825636240 P10 24 825636240 P11 24 1238454360 P12 24 137606040 P13 26 1524251520 P14 78 508083840 P15 80 743072616 P16 104 381062880 P17 121 393030144 Table 7. Conjugacy classes of maximal subgroups of L5(3) Class Order Class Size Structure M1 1965150720 121 3 : (33 : 2).L4(3).2 M2 1965150720 121 3 : (33 : 2).L4(3).2 M3 196515072 1210 36.Q8.S3.L3(3) M4 196515072 1210 36.Q8.S3.L3(3) M5 51840 4586868 C2(3).2 M6 7920 30023136 M11 M7 7920 30023136 M11 M8 605 393030144 121 : 5 155 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 Table 8. Matrix A′ for L5(3) M1 M2 M3 M4 M5 M6 M7 P1 5 5 9 9 0 0 0 P2 5 5 8 8 36 0 0 P3 2 2 3 3 0 9 9 P4 1 1 2 2 0 8 8 P5 1 1 2 2 8 0 0 P6 1 1 1 1 9 0 0 P7 2 2 3 3 0 0 0 P8 2 2 2 2 0 0 0 P9 1 1 2 2 0 0 0 P10 1 1 1 1 0 0 0 P11 2 2 3 3 0 0 0 P12 4 4 5 5 0 0 0 P13 2 2 1 1 0 0 0 P14 1 1 1 1 0 0 0 P15 1 1 0 0 0 0 0 P16 0 0 1 1 0 0 0 Observe that C = M1 ∪M3 ∪M8 is a cover of size 393031475, and therefore σ(L5(3)) ≤ 393031475. On the other hand, it is necessary to use all 393030144 subgroups from class M8 in order to cover the cyclic subgroups of order 121, and we must use at least 121 subgroups from M1 ∪M2 to cover the members of P15 and at least 1210 subgroups from M3 ∪M4 to cover the members of P16. Consequently σ(L5(3)) = 393031475. 4.2. (A5 ×A5 ×A5).6 The group (A5 × A5 × A5).6 has 19 conjugacy classes of principal subgroups and 6 classes of maximal subgroups, described in Tables 9 and 10. We note that each member of P17 is contained in a unique maximal subgroup, which is from M5. Therefore it is necessary to use all 1000 subgroups from M5 in the cover. Likewise, we must use the subgroup from M1 in order to cover the members of P8 ∪P9 ∪P10 ∪P11. The only classes of principal subgroups whose members are not covered by M1 ∪M5 are P12 and P15. We present the matrix A′ in Table 11. 156 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 Table 9. Conjugacy classes of principal subgroups of (A5 ×A5 ×A5).6 Class Order Class Size P1 5 2592 P2 6 3000 P3 6 6000 P4 6 36000 P5 10 3240 P6 10 3240 P7 10 4050 P8 12 4500 P9 12 4500 P10 12 9000 P11 12 13500 P12 12 54000 P13 15 2160 P14 15 3600 P15 15 21600 P16 15 2160 P17 18 24000 P18 30 2700 P19 30 2700 Table 10. Conjugacy classes of maximal subgroups of (A5 ×A5 ×A5).6 Class Order Class Size Structure M1 432000 1 A5.A5.S5 M2 648000 1 A5 wr 3 M3 10368 125 26.33.6 M4 6000 216 53.A4.4 M5 1296 1000 33.(22 ×A4) M6 360 3600 3 ×S5 Table 11. Matrix A′ for (A5 ×A5 ×A5).6 M2 M3 M4 M6 P12 0 1 2 1 P15 1 0 1 1 We can cover the remaining principal subgroups with the 126 subgroups from M2 ∪M3. Moreover, covering the members of P15 without using the subgroup from M2 requires at least 216 subgroups from 157 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 M4 ∪M6, and will not yield a minimal cover. Therefore we must use the subgroup from M2, and we can complete a minimal cover by finding a minimal subset of M3 ∪M4 ∪M6 which covers the elements of P12. Therefore we construct the 54000 × 3941 (0,1)-incidence matrix between the members of P12 and those of M3 ∪M4 ∪M6. We use Gurobi to solve the corresponding linear programming problem, and find that the minimum number of subgroups required to cover the members of P12 is indeed 125. Therefore M1∪M2∪M3∪M5 is a minimal cover of (A5×A5×A5).6, and σ((A5×A5×A5).6) = 1127. 4.3. The remaining groups from Tables 4 and 5 For each of the remaining groups in Tables 4 and 5 we describe a cover which is either minimal (in the case of groups from Table 4) or else has size equal to the upper bound given in Table 5. In this section Dn denotes the dihedral group of order 2n. PΣL2(121): There is a minimal cover C (of size 794) which is a union of three conjugacy classes of maximal subgroups. C consists of the normal subgroup L2(121), the conjugacy class of subgroups isomorphic to 112 : (5 × D12) (122 subgroups), and all 671 subgroups from one of the two classes of subgroups isomorphic to 2 ×L2(11).2 (and no subgroups from the other class). A5 wr 3: There is a minimal cover C (of size 317) consisting of the following subgroups: the normal subgroup A35, all 216 subgroups from the conjugacy class of maximal subgroups isomorphic to D5 wr 3, and 100 of the 125 subgroups (all conjugate) which are isomorphic to 26 : 3 : 31+2+ . PΓL2(125): There is a minimal cover C (of size 7876) which is the union of two conjugacy classes of maximal subgroups, namely the class of 7750 subgroups isomorphic to 7 : (2 ×D9 : 3), and the class of 126 subgroups isomorphic to 53 : (4 × 31 : 3). L7(2): There is a minimal cover C (of size 184308218125) which is the union four conjugacy classes of maximal subgroups, namely the class of 184308203520 subgroups isomorphic to 127 : 7, one of the two classes of 127 subgroups isomorphic to 26 : L6(2), one of the two classes of 2667 subgroups isomorphic to 210.S3.L5(2), and one of the two classes of 11811 subgroups isomorphic to 212.A8.L2(7). A7 wr 2: There is a cover of size 667 which is the union of three conjugacy classes of maximal subgroups, namely the class containing the normal subgroup A27, one of the two conjugacy classes of subgroups isomorphic to L2(7) wr 2 (225 subgroups), and the conjugacy class of 441 subgroups isomorphic to S5 wr 2. L3(4).22: There is a cover C of size 166 consisting of the following maximal subgroups: the normal subgroup L3(4), all 105 subgroups isomorphic to 42.S4 (all conjugate), and a total of 60 subgroups isomorphic to SO3(7). There are three conjugacy classes of maximal subgroups isomorphic to SO3(7) in L3(4).22, each of size 120. The cover C contains 28 subgroups from one of these classes, 32 subgroups from a second class, and no subgroups from the remaining class. HS : 2: There is a cover of size 22375 which is the union of four conjugacy classes of maximal subgroups: the class of 100 subgroups isomorphic to M22.2, the class of 1100 subgroups isomorphic to 2 ×S8, the class of 5775 subgroups isomorphic to 4.25.S5, and the class of 15400 subgroups isomorphic to 22.A6.22. L2(11) wr 2: There is a cover C of size 817 consisting of the normal subgroup L2(11)2, a total of 77 subgroups isomorphic to A5 wr 2 (66 from one conjugacy class of maximal subgroups isomorphic to A5 wr 2 and 11 subgroups from the other), all 144 subgroups from the conjugacy class of maximal subgroups isomorphic to (11 : 5) wr 2, and 595 subgroups from the conjugacy class of 660 maximal subgroups isomorphic to PGL2(11). PGU3(5): There is a cover of size 6378 consisting of the following subgroups: all 6000 maximal subgroups isomorphic to 3 × 7 : 3 (all conjugate), all 126 maximal subgroups isomorphic to 51+2+ : 24 (all conjugate), and 252 maximal subgroups isomorphic to 3 × SL2(5) : 2 (out of 525, all of which are conjugate). 158 M. Epstein / J. Algebra Comb. Discrete Appl. 9(3) (2022) 149–159 References [1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265. [2] J. R. Britnell, A. Evseev, R. M. Guralnick, P. E. 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