# On Some Combinatorial Structures Constructed from the GroupsL(3,5), U(5,2), and S(6,2)

Authors: Dean Crnković; Vedrana Mikulić Crnković
Journal: International Journal of Combinatorics
 (2011)

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## Abstract

We describe a construction of primitive 2-designs and strongly regular graphs from the simple groupsL(3,5),U(5,2), and S(6,2). The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of L(3,5),U(5,2), and S(6,2). In addition, from the group S(6,2), we construct 2-designs with parameters (28,4,4) and (28,4,1) having the full automorphism group isomorphic to U(3,3):Z2.


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## Body

## 1. Introduction

Questions about combinatorial structures related to finite groups arose naturaly in studying of the groups. Studies on the interplay between finite groups and combinatorial structures have provided many useful and interesting results. From a geometric point of view, the most interesting designs are generally those admitting large automorphism groups. Famous Witt designs constructed from Mathieu groups have been discovered in 1930’s (see [1, 2]), and for some further construction of combinatorial structures from finite groups, we refere the reader to [3–5]. A construction of 1-designs and regular graphs from primitive groups is described in [6] and corrected in [7]. The construction employed in this paper is described in [8], as a generalization of the one from [6, 7].An incidence structure is an ordered triple𝒟=(𝒫,ℬ,ℐ) where 𝒫 and ℬ are nonempty disjoint sets and ℐ⊆𝒫×ℬ. The elements of the set 𝒫 are called points, the elements of the set ℬ are called blocks, and ℐ is called an incidence relation. If |𝒫|=|ℬ|, then the incidence structure is called symmetric. The incidence matrix of an incidence structure is a b×v matrix [mij] where b and v are the number of blocks and points, respectively, such that mij=1 if the point Pj and block xi are incident, and mij=0 otherwise. An isomorphism from one incidence structure to another is a bijective mapping of points to points and blocks to blocks which preserves incidence. An isomorphism from an incidence structure 𝒟 onto itself is called an automorphism of 𝒟. The set of all automorphisms forms a group called the full automorphism group of 𝒟 and is denoted by Aut(𝒟).At-(v,k,λ) design is a finite incidence structure (𝒫,ℬ,ℐ) satisfying the following requirements: (1)
|𝒫|=v,(2)
every element ofℬ is incident with exactly k elements of 𝒫,(3)
everyt elements of 𝒫 are incident with exactly λ elements of ℬ.A2-(v,k,λ) design is called a block design. A 2-(v,k,λ) design is called quasisymmetric if the number of points in the intersection of any two blocks takes only two values. If |𝒫|=|ℬ|, then the design is called symmetric. A symmetric 2-(v,k,1) design is called a projective plane.An incidence structure has repeated blocks if there are two blocks incident with exactly the same points. An incidence structure that has no repeated blocks is called simple. All designs described in this paper are simple.Let𝒢=(𝒱,ℰ,ℐ) be a finite incidence structure. 𝒢 is a graph if each element of ℰ is incident with exactly two elements of 𝒱. The elements of 𝒱 are called vertices, and the elements of ℰ are called edges. Two vertices u and v are called adjacent or neighbors if they are incident with the same edge. The number of neighbors of a vertex v is called the degree of v. If all the vertices of the graph 𝒢 have the same degree k, then 𝒢 is called k-regular. Define a square {0,1}-matrix A=(auv) labelled with the vertices of 𝒢 in such a way that auv=1 if and only if the vertices u and v are adjacent. The matrix A is called the adjacency matrix of the graph 𝒢.A graph𝒢 is called a strongly regular graph with parameters (n,k,λ,μ) and denoted by SRG(n,k,λ,μ) if 𝒢 is k-regular graph with n vertices and if any two adjacent vertices have λ common neighbors and any two nonadjacent vertices have μ common neighbors.Letx and y(x<y) be the two cardinalities of block intersections in a quasi-symmetric design 𝒟. The block graph of 𝒟 has as vertices the blocks of 𝒟 and two vertices are adjacent if and only if they intersect in y points. The block graph of a quasi-symmetric 2-(v,k,λ) design is strongly regular. In a 2-(v,k,1) design which is not a projective plane, two blocks intersect in 0 or 1 points; therefore, the block graph of this design is strongly regular (see [9]).Let𝒟 be a symmetric (v,k,λ) design which possesses a symmetric incidence matrix M with 1 everywhere on the diagonal. Then, the matrix M-I is an adjacency matrix of a strongly regular graph 𝒢 with parameters (v,k-1,λ-2,λ) (see [9]) and Aut(𝒢)≤Aut(𝒟).LetG be a simple group and H be a maximal subgroup of G. The conjugacy class of H is denoted by cclG(H). Obviously, NG(H)=H, so |cclG(H)|=[G:H]. Denote the elements of the conjugacy class cclG(H) by Hg1,Hg2,…,Hgj, j=[G:H].In this paper, we consider block designs constructed from the linear groupL(3,5), strongly regular graphs constructed from the unitary group U(5,2), and block designs and strongly regular graphs constructed from the symplectic group S(6,2).L(3,5) is the simple group of order 372000, and it has five distinct classes of maximal subgroups: H1≅H2≅(Z5×Z5):GL(2,5), H3≅S5, H4≅(Z4×Z4):S3, and H5≅F93. We define incidence structures on the elements of conjugacy classes of the maximal subgroups of L(3,5); that is, points and blocks are labelled by elements of conjugacy classes of the maximal subgroups of L(3,5).U(5,2) is the simple group of order 13685760, and it has six distinct classes of maximal subgroups: K1≅(E64:Z2)·(E9:Z3)·SL(2,3), K2≅Z3×U(4,2), K3≅(E16:E16):(Z3×A5), K4≅E81:S5, K5≅(S3×(E9:Z3)):SL(2,3), and K6≅L(2,11). We define strongly regular graphs whose vertices are labelled by elements of conjugacy classes of the maximal subgroups of U(5,2).S(6,2) is the simple group of order 1451520, and it has eight distinct classes of maximal subgroups: M1≅U(4,2):Z2, M2≅S8, M3≅E32:S6, M4≅U(3,3):Z2, M5≅E64:L(3,2), M6≅((E16:Z2)×E4):(S4×S4), M7≅S3×S6, and M8≅L(2,8):Z3. We define incidence structures on the elements of conjugacy classes of the maximal subgroups of S(6,2) and strongly regular graphs whose vertices are labelled by elements of conjugacy classes of the maximal subgroups of S(6,2).Generators of groupsL(3,5), U(5,2), and S(6,2) and their maximal subgroups are available on the Internet: http://brauer.maths.qmul.ac.uk/Atlas/.In this paper, we describe a construction of primitive block designs with parameters(28,12,11), (28,4,5), (28,10,40), (31,6,1), (31,6,100), (31,10,300), (31,15,700), (31,3,25), (31,12,550), (31,15,875), (36,16,12), (36,8,6), (36,12,33), (36,6,8), and (63,31,15) and strongly regular graphs with parameters (63,30,13,15), (120,56,28,24), (135,64,28,24), (165,36,3,9), (176,40,12,8), (297,40,7,5), and (1408,567,246,216). The designs and the graphs are constructed by defining incidence structures on conjugacy classes of maximal subgroups of the simple groups L(3,5),U(5,2), and S(6,2). In addition, from the group S(6,2), we construct 2-designs with parameters (28,4,4) and (28,4,1) having the full automorphism group isomorphic to U(3,3):Z2.The graphs described in this paper have been previously known, since they can be constructed as rank 3 graphs. For more details on rank 3 graphs, we refer the reader to [9]. Some of the constructed block designs on 31 points have large number of blocks, and we did not find a record that they have been previously studied, although these designs can be constructed from PG(2,5). The constructed designs with parameters (28,12,11), (28,4,5), (36,16,12), (36,8,6), and (63,31,15) are isomorphic to the already known designs. However, we did not find an evidence that the constructed designs with parameters (28,10,40), (36,12,33), and (36,6,8) are isomorphic to the already known objects.The paper is organized as follows: in Section2, we describe the method of construction of primitive designs and graphs used in this paper, Section 3 describes block designs on 31 points constructed from the group L(3,5), Section 4 gives strongly regular graphs constructed from the group U(5,2), and Section 5 describes the group S(6,2) as block designs and strongly regular graphs. At the end of the paper, we give a list of the constructed designs and strongly regular graphs and their full automorphism groups.For basic definitions and group theoretical notation, we refer the reader to [10, 11].


## 2. The Construction

The following construction of symmetric 1-designs and regular graphs is presented in [6, 12].Theorem 2.1.
Let G  be a finite primitive permutation group acting on the setΩ of size n. Let α∈Ω, and let Δ≠{α} be an orbit of the stabilizer Gα of α. If ℬ={Δg:g∈G} and, given δ∈Δ, ℰ={{α,δ}g:g∈G}, then 𝒟=(Ω,ℬ) forms a symmetric 1-(n,|Δ|,|Δ|) design. Further, if Δ is a self-paired orbit of Gα, then Γ(Ω,ℰ) is a regular connected graph of valency |Δ|, 𝒟 is self-dual, and G acts as an automorphism group on each of these structures, primitive on vertices of the graph and on points and blocks of the design.In [8], we introduced a generalization of the above construction. This generalization, presented below in Theorem 2.2, allows us to construct 1-designs that are not necessarily symmetric and stabilizers of a point and a block are not necessarily conjugate.Theorem 2.2.
LetG be a finite permutation group acting primitively on the sets Ω1 and Ω2 of size m and n, respectively. Let α∈Ω1, δ∈Ω2, and let Δ2=δGα be the Gα-orbit of δ∈Ω2 and Δ1=αGδ be the Gδ-orbit of α∈Ω1. If Δ2≠Ω2 and
(2.1)B={Δ2g:g∈G},
then 𝒟(G,α,δ)=(Ω2,ℬ) is a 1-(n,|Δ2|,|Δ1|) design with m  blocks and G acts as an automorphism group, primitive on points and blocks of the design.The construction of a design described in Theorem2.2 can be interpreted in the following way: (i)
the point set isΩ2=δG,(ii)
the block set isΩ1=αG,(iii)
the blockαg′ is incident with the set of points {δg:g∈Gαg′}.Let a pointδg∈Ω2 be incident with a block αg′∈Ω1. Then, g∈Gαg′; hence, there exists g¯∈Gα such that g=g¯g′. Therefore,
(2.2)Gαg′∩Gδg=Gαg′∩Gδg¯g′=Gαg′∩Gδg¯g′=(Gα∩Gδg¯)g′=(Gα∩Gδg¯)g′=(Gαg¯-1∩Gδ)g¯g′=(Gα∩Gδ)g¯g′=(Gα∩Gδ)g.If a pointδg∈Ω2 is incident with the block α∈Ω1, then Gα∩Gδg=(Gα∩Gδ)g. If the set {Gα∩Gδg∣g∈G} contains Orb(Gα,Ω2) Gα-conjugacy classes, where Orb(Gα,Ω2) is the number of Gα-orbits on Ω2, then each conjugacy class corresponds to one Gα-orbit and the incidence relation in the design 𝒟(G,α,δ) can be defined as follows:(i) the blockαg′ is incident with the point δg if and only if Gαg′∩Gδg is conjugate to Gα∩Gδ.Similarly, if the set{Gα∩Gδg∣g∈G} contains Orb(Gα,Ω2) isomorphism classes, then the incidence in the design 𝒟(G,α,δ) can be defined as follows:(ii) the blockαg′ is incident with the point δg if and only if Gαg′∩Gδg≅Gα∩Gδ.In the construction of the design𝒟(G,α,δ) described in Theorem 2.2, instead of taking a single Gα-orbit, we can take Δ2 to be any union of Gα-orbits.Corollary 2.3.
LetG be a finite permutation group acting primitively on the sets Ω1 and Ω2 of size m and n, respectively. Let α∈Ω1 and Δ2=⋃i=1sδiGα, where δ1,…,δs∈Ω2 are representatives of distinct Gα-orbits. If Δ2≠Ω2 and
(2.3)B={Δ2g:g∈G},
then 𝒟(G,α,δ1,…,δs)=(Ω2,ℬ) is a 1-(n,|Δ2|,∑i=1s|αGδi|) design with m  blocks and G acts as an automorphism group, primitive on points and blocks of the design.Proof.
Clearly, the number of points isv=n, since the point set is 𝒫=Ω2. Further, each element of ℬ consists of k=|Δ2| elements of Ω2.
The setΔ2 is a union of Gα-orbits, so Gα⊆GΔ2, where GΔ2 is the setwise stabilizer of Δ2. Since G is primitive on Ω1, Gα is a maximal subgroup of G, and therefore GΔ2=Gα. The number of blocks is(2.4)b=|Δ2G|=|G||GΔ2|=|G||Gα|=|Ω1|=m.
SinceG acts transitively on Ω1 and Ω2, the constructed structure is a 1-design, hence bk=vr, where each point is incident with r blocks. Therefore,(2.5)|Ω1||Δ2|=|Ω2|r,
and consequently
(2.6)|G||Gα|∑i=1s|Gα||(Gα)δi|=|G||Gδ1|r.
It follows that
(2.7)r=∑i=1s|Gδ1||(Gα)δi|=∑i=1s|Gδi||(Gδi)α|=∑i=1s|αGδi|.Remark 2.4.
In the construction of graphs described in Theorem2.1, we can define the set of edges ℰ as a union ⋃i=1s{{α,δi}g:g∈G}.The construction described in Corollary2.3 gives us all designs that admit primitive action of the group G on points and blocks.Corollary 2.5.
If the groupG acts primitively on the points and the blocks of a 1-design 𝒟, then 𝒟 can be obtained as described in Corollary 2.3, that is, such that Δ2 is a union of Gα-orbits.Proof.
Letα be any block of the design 𝒟. G acts transitively on the block set ℬ of the design 𝒟, hence ℬ=αG. Since G acts primitively on ℬ, the stabilizer Gα is a maximal subgroup of G. Gα fixes α, so α is a union of Gα-orbits.We can obtain a 1-design by defining the incidence in such a way that the pointδg is incident with the block αg′ if and only if
(2.8)Gαg′∩Gδg≅Gi,i=1,…,k,
where {G1,…,Gk}⊂{Gαx∩Gδy∣x,y∈G}.LetG be a simple group and H1 and H2 be maximal subgroups of G. The stabilizer of any element Hix∈cclG(Hi), i=1,2, is the maximal subgroup Hix, hence G acts primitively on the class cclG(Hi), i=1,2, by conjugation and
(2.9)|cclG(H1)|=[G:H1]=m,|cclG(H2)|=[G:H2]=n.Corollary2.3 allows us to define a 1-design on conjugacy classes of the maximal subgroups H1 and H2 of a simple group G. Let us denote the elements of cclG(H1) by H1g1,H1g2,…,H1gm and the elements of cclG(H2) by H2h1,H2h2,…,H2hn.We can construct a 1-design such that(i)
the point set of the design iscclG(H2),(ii)
the block set iscclG(H1),(iii)
the blockH1gi is incident with the point H2hj if and only if H2hj∩H1gi≅Gi,  i=1,…,k, where {G1,…,Gk}⊂{H2x∩H1y∣x,y∈G}.Let us denote a 1-design constructed in this way by𝒟(G,H2,H1;G1,…,Gk).Similarly, from the conjugacy class of a maximal subgroupH of a simple group G, one can construct regular graph in the following way:  (i)
the vertex set of the graph iscclG(H),(ii)
the vertexHgi is adjacent to the vertex Hgj if and only if Hgi∩Hgj≅Gi, i=1,…,k, where {G1,…,Gk}⊂{Hx∩Hy∣x,y∈G}.We denote a regular graph constructed in this way by𝒢(G,H;G1,…,Gk).Remark 2.6.
Letϕ be an automorphism of a finite group G. Then, the design 𝒟(G,H2,H1;G1,…,Gk) is isomorphic to 𝒟(G,(H2)ϕ,(H1)ϕ;G1,…,Gk), and the graph 𝒢(G,H;G1,…,Gk) is isomorphic to 𝒢(G,(H)ϕ;G1,…,Gk).


## 3. Block Designs on 31 Points Constructed from the GroupL(3,5)

LetG be a group isomorphic to the linear group L(3,5).Using GAP (see [13]), one can check that H2x∩H1y≅(Z2·A5):Z4 or ((Z5×Z5)·Z5):(Z4×Z4) for all x,y∈G. Further, for every H2x, (3.1)|{H1y∣y∈G,H1y∩H2x≅((Z5×Z5)⋅Z5):(Z4×Z4)}|=6.Let us define setsSi(1)={H2gj∈cclG(H2)∣H1hi∩H2gj≅((Z5×Z5)·Z5):(Z4×Z4)}, 1≤i≤31. For every 1≤i, k≤31, i≠k, the set Si(1)∩Sk(1) has exactly one element. That proves that the incidence structure 𝒟1=𝒟(L(3,5),H1,H2;((Z5×Z5)·Z5):(Z4×Z4)) is the unique symmetric 2-(31,6,1) design (see [14]), that is, the projective plane PG(2,5). The full automorphism group of the design 𝒟1 has 372000 elements, and it is isomorphic to the group L(3,5).LetH3≅S5 be a maximal subgroup in G. The conjugacy class cclG(H3) has 3100 elements. One can verify using GAP that H1x∩H3y is isomorphic to D8, D12, or Z5:Z4, for all x,y∈G. Using GAP (see [13]), we obtain the following:(1)
|{H1hi∈cclG(H1)∣H1hi∩H3gj≅Z5:Z4}|=6, for all H3gj∈cclG(H3), 1≤j≤3100,(2)
|Si(2)∩Sk(2)|=100, for all i,k∈{1,2,…,31}, i≠j, Si(2)={H3gjcclG(H3)∣H1hi∩H3gj≅Z5:Z4},(3)
|{H1hi∈cclG(H1)∣H1hi∩H3gj≅D12}|=10, for all H3gj∈cclG(H3), 1≤j≤3100,(4)
|Si(3)∩Sk(3)|=300, for all i,k∈{1,2,…,31}, i≠j, Si(3)={H3gj∈cclG(H3)∣H1hi∩H3gj≅D12},(5)
|{H1hi∈cclG(H1)∣H1hi∩H3gj≅D8}|=15, for all H3gj∈cclG(H3), 1≤j≤3100,(6)
|Si(4)∩Sk(4)|=700, for all i,k∈{1,2,…,31}, i≠j, Si(4)={H3gj∈cclG(H3)∣H1hi∩H3gj≅D8}.It follows from (1) and (2) that the structure𝒟2=𝒟(L(3,5),H1,H3;Z5:Z4) is a block design 2-(31,6,100). The blocks of 𝒟2 are conics in PG(2,5).(3) and (4) imply that the structure𝒟3=𝒟(L(3,5),H1,H3;D12) is a block design 2-(31,10,300). The blocks of 𝒟3 are interior points of conics in PG(2,5).Further, (5) and (6) show that the structure𝒟4=𝒟(L(3,5),H1,H3;D8) is a block design 2-(31,15,700). The blocks of 𝒟3 are exterior points of conics in PG(2,5).The full automorphism group of designs𝒟2, 𝒟3, and 𝒟4 has 372000 elements and is isomorphic to the group L(3,5).LetH4≅(Z4×Z4):S3 be a maximal subgroup in G. The conjugacy class cclG(H4) has 3875 elements. Using GAP (see [13]), one can check that H1x∩H4y is isomorphic to S3, Z2×Z4, or (Z8:Z2):Z2, for all x,y∈G. In the similar way as above, we obtain the following results:(i)
𝒟5=𝒟(L(3,5),H1,H4;(Z8:Z2):Z2)) is a 2-(31,3,25) design. 𝒟5 is isomorphic to the design constructed from the symmetric 2-(31,6,1) design 𝒟1 in such a way that the blocks are all triples of noncollinear points.(ii)
The incidence structure𝒟6=𝒟(L(3,5),H1,H4;Z2×Z4) is a block design 2-(31,12,550). The blocks of 𝒟6 are unions of sides of triangles in PG(2,5).(iii)
𝒟7=𝒟(L(3,5),H1,H4;S3) is a block design 2-(31,15,875). The blocks of 𝒟7 are unions of sides of triangles in PG(2,5), including the corners.The full automorphism group of𝒟5, 𝒟6, and 𝒟7 is isomorphic to L(3,5).Six designs isomorphic to𝒟2, 𝒟3, 𝒟4, 𝒟5, 𝒟6, and 𝒟7 can be obtain in the same way as described in this paper using the maximal subgroup H2 instead of H1. This is a consequence of the Remark 2.6 and the fact that there exists an automorphism of L(3,5) which fixes cclG(H3), cclG(H4) and cclG(H5), setwise and acts as a transposition which maps cclG(H1) onto cclG(H2).We thank an anonymous referee of an earlier version of this paper for suggesting the construction of the designs𝒟2, 𝒟3, 𝒟4, 𝒟6, and 𝒟7 from triangles and conics in PG(2,5).


## 4. Strongly Regular Graphs Constructed from the GroupU(5,2)

In this section, we describe structures constructed from a simple group isomorphic to the unitary groupU(5,2).The intersection of two different elementsKix,Kiy∈cclG(Ki), i=1,2,3,4, denoted by Gi,ik, is isomorphic to (i)
G1,11≅((Ex27+:Z2)·Z4):Z3 or G1,12≅((E8×Z4)·E8):E9,(ii)
G2,21≅(E8:Z12):Z6 or G2,22≅(((Ex27+:Z2)·Z4):Z3):Z3,(iii)
G3,31≅A5×Z3 or G3,32≅Ex128-:E9,(iv)
G4,41≅D6, G4,42≅S3×S3, G4,43≅(A4:Z2):Z3, G4,44≅A5:Z2, or G4,45≅(Ex27+:Z2)×S3.Applying the method described in Section2, we obtain the following results:  (i)
the graph𝒢1=𝒢(U(5,2),K1;G1,11) is a strongly regular graph with parameters (165,36,3,9),(ii)
the graph𝒢2=𝒢(U(5,2),K2;G2,22) is a strongly regular graph with parameters (176,40,12,8),(iii)
the graph𝒢3=𝒢(U(5,2),K3;G3,32) is a strongly regular graph with parameters (297,40,7,5),(iv)
the graph𝒢4=𝒢(U(5,2),K4;G4,42,G4,43,G4,44) is a strongly regular graph with parameters (1408,567,246,216).The full automorphism groups of the graphs𝒢1, 𝒢2, and 𝒢3 are isomorphic to the group U(5,2):Z2≅Aut(U(5,2)). U(5,2) is a rank 3 group on 165, 176, and 297 points, and 𝒢1, 𝒢2, and 𝒢3 are rank 3 graphs of the group U(5,2).The full automorphism group of the graph𝒢4 is of order 18393661440 and is isomorphic to the group U(6,2):Z2≅Fi21:Z2. Although U(5,2) acts as a rank 7 group on 1408 points, the graph 𝒢4 can be constructed as a rank 3 graph from the Fischer group Fi21.


## 5. Structures Constructed from the GroupS(6,2)

In this section, we consider the structures constructed from a simple group isomorphic to the symplectic groupS(6,2).The intersection of two different elementsMix∈cclG(Mi) and Mjy∈cclG(Mj), denoted by Gi,jk, is isomorphic to (we introduce only the intersection of elements of conjugacy classes that give rise to a strongly regular graph or a block design)  (i)
G1,31≅E16:S5 or G1,32≅A6:E4,(ii)
G1,61≅((E8·Z12):Z6):Z2 or G1,62≅S4×D8,(iii)
G1,71≅E27:(D8×Z2) or G1,72≅S5:Z2,(iv)
G2,31≅A6:E4 or G2,32≅(S4×S4):Z2,(v)
G2,51≅E8:L(2,7) or G2,52≅E16:S4,(vi)
G2,61≅E16:S4 or G2,62≅S4×D8,(vii)
G2,71≅S5×S3 or G2,72≅E9:(D8×Z2),(viii)
G3,31≅Ex32+:(D12×Z2) or G3,32≅S6,(ix)
G4,41≅(Ex27+:Z2)·Z4 or G4,42≅(Z4×Z4):D12,(x)
G5,51≅L(2,7), G5,52≅E16:D12, or G5,53≅Ex32+:(A4:Z2).Applying the method described in Section2, we obtain the following results:  (i)
the incidence structure𝒟8=𝒟(S(6,2),M1,M3;G1,31) is a block design with parameters 2-(28,12,11),(ii)
the incidence structure𝒟9=𝒟(S(6,2),M1,M6;G1,61) is a block design with parameters 2-(28,4,5),(iii)
the incidence structure𝒟10=𝒟(S(6,2),M1,M7;G1,71) is a block design with parameters 2-(28,10,40),(iv)
the incidence structure𝒟11=𝒟(S(6,2),M2,M3;G2,31) is a block design with parameters 2-(36,16,12),(v)
the incidence structure𝒟12=𝒟(S(6,2),M2,M5;G2,51) is a block design with parameters 2-(36,8,6),(vi)
the incidence structure𝒟13=𝒟(S(6,2),M2,M6;G2,61) is a block design with parameters 2-(36,12,33),(vii)
the incidence structure𝒟14=𝒟(S(6,2),M2,M7;G2,71) is a block design with parameters 2-(36,6,8),(viii)
the incidence structure𝒟15=𝒟(S(6,2),M3,M3;G3,31,M3) is a block design with parameters 2-(63,31,15),(ix)
the graph𝒢5=𝒢(S(6,2),M3;G3,31) is a strongly regular graph with parameters (63,30,13,15),(x)
the graph𝒢6=𝒢(S(6,2),M4;G4,41) is a strongly regular graph with parameters (120,56,28,24),(xi)
the graph𝒢7=𝒢(S(6,2),M5;G5,51) is a strongly regular graph with parameters (135,64,28,32).The full automorphism groups of designs𝒟8,𝒟9,𝒟10, 𝒟11,𝒟12,𝒟13, and 𝒟14 and the full automorphism group of the graph 𝒢5 are isomorphic to S(6,2).Graphs𝒢6 and 𝒢7 have O8+(2):Z2≅AutO8+(2) as the full automorphism group. O8+(2) acts as a rank 3 group on 120 and 135 points, and graphs 𝒢6 and 𝒢7 can be constructed as rank 3 graphs from the orthogonal group O8+(2). On the other hand, the group S(6,2) acts as a rank 3 group on 120 points and as a rank 4 group on 135 points. Therefore, the graph 𝒢6 can also be constructed as a rank 3 graph from the S(6,2).The design𝒟8 is a quasi-symmetric SDP design isomorphic to the design described in [15, 16]. Its block graph is isomorphic to the complement of the graph 𝒢5.Furthermore, from the design𝒟9 on 28 points, one can construct two designs on 28 points, one isomorphic to the design 𝒟8 and one isomorphic to the design 𝒟10. The design on 28 points, whose blocks are unions of three disjoint blocks of the design 𝒟9 such that stabilizer of that union under the action of the automorphism group Aut(𝒟9) is isomorphic to the maximal subgroup M3, is isomorphic to the design 𝒟8. On the other hand, unions of three blocks intersecting in exactly one point of the design 𝒟9 such that stabilizer of that union under the action of the group Aut(𝒟9) is isomorphic to the maximal subgroup M7 are blocks of a design isomorphic to the design 𝒟10.The design𝒟11 is also a quasi-symmetric design, isomorphic to the design described in [15, 16]. Its block graph is isomorphic to the graph 𝒢5.The ovals (sets of 6 points, no three collinear) of the design𝒟12 form a block design with parameters 2-(36,6,8) isomorphic to the design 𝒟14. Furthermore, the group S(6,2) has a subgroup H isomorphic to the group PG(2,8). The group H acts on the set of all ovals of the design 𝒟12 in four orbits of size 84 and one of these orbits forms the block set of a block design with parameters 2-(36,6,2) having Aut(PGL(2,8)) as the full automorphism group. The design 𝒟12 is isomorphic to the 2-(36,8,6) design described in [17].Unions of four disjoint blocks of the design𝒟14 such that stabilizer of that union under the action of the automorphism group Aut(𝒟14) is isomorphic to the maximal subgroup M6 are blocks of a design isomorphic to the complement of the design 𝒟13.The design𝒟15 is the point-hyperplane design in the projective geometry PG(6,2), and its full automorphism group is isomorphic to PG(6,2). 𝒟15 possesses a symmetric incidence matrix with 1 everywhere on the diagonal, and therefore it gives rise to a strongly regular graph with parameters (63,30,13,15) which is isomorphic to the graph 𝒢5.The graph𝒢7 can also be constructed from the design 𝒟12. Any two blocks of 𝒟12 intersect in 1, 2, or 4 points. The graph which has as its vertices the blocks of 𝒟12, two vertices being adjacent if and only if the corresponding blocks intersect in one point, is isomorphic to 𝒢7.LetU≅U(3,3) be a subgroup of the group S(6,2) and let cclU(M1) and cclU(M6) be conjugacy classes of maximal subgroups M1 and M6 under the action of the group U. The intersection of M1g, g∈U, and M6h, h∈U, is isomorphic either to G1,61 or G1,62. Let 𝒟′ be an incidence structure whose points are labeled by the elements of the class cclU(M1) and whose blocks are labeled by the elements of the class cclU(M6), point and block being incident if and only if the intersection of corresponding elements of conjugacy classes is isomorphic to the group G1,61. The structure 𝒟′ is isomorphic to the block design with parameters 2-(28,4,4) having AutU(3,3)≅U(3,3):Z2 as the full automorphism group. Further, let M=M6g,g∉U, be an element of the class cclS(6,2)(M6). The incidence structure 𝒟′′ whose points are labeled by the elements of the class cclU(M1) and whose blocks are labeled by the elements of the class cclU(M), point and block being incident if and only if the intersection of corresponding elements of conjugacy classes is isomorphic to the group G1,61, is isomorphic to the Hermitian unital with parameters 2-(28,4,1) having AutU(3,3) as the full automorphism group. We conclude that the designs 𝒟′ and 𝒟′′ are subdesigns of the design 𝒟9. One can construct from the group U(3,3) the design 𝒟(U(3,3),(E9:Z3):Z8,Z4·S4;Z3:Z8) isomorphic to the design 𝒟′′ (see [15]).The Hölz designH(q) of order q is a design with parameters 2-(q3+1,q+1,q+2) which is a union of the Hermitian unital with parameters 2-(q3+1,q+1,1) and a 2-(q3+1,q+1,q+1) design whose blocks are arcs in the unital. To prove that design 𝒟9 is isomorphic to the design H(3), we constructed design H(3), by the method described in [18], as the support design of the dual code of the code spanned by the incidence vectors of the design 𝒟8. Constructed design is isomorphic to the design 𝒟9.From(i)
|cclS(6,2)M1|=|cclUM1|,(ii)
|cclS(6,2)M2|=|cclUM2|,(iii)
|cclS(6,2)M3|=|cclUM3|,(iv)
|cclS(6,2)M7|=|cclUM7|,one can conclude that, in order to construct designs𝒟8, 𝒟10, 𝒟11, 𝒟14 and 𝒟15, corresponding maximal subgroups need not to be conjugate by the elements of the whole group S(6,2). The conjugation by the elements of the subgroup U is sufficient to obtain the desired structures.In Table4 we give a list of the constructed designs and strongly regular graphs and their full automorphism groups. Table 1, Table 2, and Table 3 give a list of maximal subgroups of the groups L(3,5), U(5,2), and S(6,2), respectively.Table 1
Maximal subgroups of the groupL(3,5).
Maximal subgroupStructureOrderIndexH1E25:GL(2,5)1200031H2E25:GL(2,5)1200031H3S51203100H4(Z4×Z4):S3963875H5F93934000Table 2
Maximal subgroups of the groupU(5,2).
Maximal subgroupStructureOrderIndexK1(E64:Z2)·(E9:Z3)·SL(2,3)829444165K2Z3×U(4,2)46080176K3(E16:E16):(Z3×A5)77760297K4E81:S597201408K5(S3×(E9:Z3)):SL(2,3)38883520K6L(2,11)66020736Table 3
Maximal subgroups of the groupS(6,2).
Maximal subgroupStructureOrderIndexM1U(4,2):Z25184028M2S84032036M3E32:S62304063M4U(3,3):Z212096120M5E64:L(3,2)10752135M6((E16:Z2)×E4):(S4×S4)4608315M7S3×S64320336M8L(2,8):Z3504960Table 4
Structures constructed fromL(3,5), U(5,2), and S(6,2).
Combinatorial structureThe full automorphism group2-(31,6,1) designL(3,5)2-(31,6,100) designL(3,5)2-(31,10,300) designL(3,5)2-(31,15,700) designL(3,5)2-(31,3,25) designL(3,5)2-(31,12,550) designL(3,5)2-(31,15,875) designL(3,5)2-(28,12,11) designS(6,2)2-(28,4,5) designS(6,2)2-(28,10,40) designS(6,2)2-(36,16,12) designS(6,2)2-(36,8,6) designS(6,2)2-(36,12,33) designS(6,2)2-(36,6,8) designS(6,2)2-(63,31,15) designPGL(6,2)2-(28,4,4) designU(3,3):Z22-(28,4,1) designU(3,3):Z2SRG(63,30,13,15)S(6,2)SRG(120,56,28,24)O8+(2):Z2SRG(135,64,28,32)O8+(2):Z2SRG(165,36,3,9)U(5,2):Z2SRG(176,40,12,8)U(5,2):Z2SRG(297,40,7,5)U(5,2):Z2SRG(1408,567,246,216)Fi21:Z2



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*Source: 137356-2011-09-04.xml*