On rank 2 and rank 3 residually connected geometries for M 22

Transcription

1 Note di Matematica 22, n. 1, 2003, On rank 2 and rank 3 residually connected geometries for M 22 Nayil Kilic Department of Mathematics, UMIST, P.O. Box 88, Manchester, M60 1QD, UK. mcbignk3@stud.umist.ac.uk Peter Rowley Department of Mathematics, UMIST, P.O. Box 88, Manchester, M60 1QD, UK. peter.rowley@umist.ac.uk Received: 29/11/2002; accepted: 29/7/2003. Abstract. We determine all rank 2 and rank 3 residually connected flag transitive geometries for the Mathieu group M 22 for which object stabilizer are maximal subgroups. Keywords: Residually connected flag transitive geometries, Mathieu group MSC 2000 classification: 51Exx Introduction In this paper we shall describe all rank 2 and rank 3 residually connected flag transitive geometries for the Mathieu group M 22 whose object stabilizers are maximal subgroups. We begin by reviewing geometries and some standard notation. A geometry is a triple (Γ, I, ) where Γ is a set, I an index set and a symmetric incidence relation on Γ which satisfy (i) Γ =. i I Γ i ; and (ii) if x Γ i, y Γ j (i, j I) and x y, then i j. The elements of Γ i are called objects of type i, and I is the rank of the geometry Γ (as is usual we use Γ is place of the triple (Γ, I, )). A flag F of Γ is a subset of Γ which, for all x, y F, x y, x y. The rank of F is F, the corank of F is I\F and the type of F is {i I F Γ i }. All geometries we consider are assumed to contain at least one flag of rank I. The automorphism group of Γ, AutΓ, consists of all permutations of Γ which preserve the sets Γ i and the incidence relation. Let G be a subgroup of AutΓ. We call Γ a flag transitive geometry for G if for any two flags F 1 and F 2 of Γ having the same type, there exists g G such that F 1 g = F 2. For Γ, the residue of, denoted Γ, is defined to be {x Γ x y for all y }. A geometry Γ is called residually connected if for all flags F of Γ of corank 2 the incidence graph of

2 108 N. Kilic, P. Rowley Γ F is connected. Now suppose that Γ is a flag transitive geometry for the group G. As is well-known we may view Γ in terms of certain cosets of G. This is the approach we shall follow here. For each i I choose an x i Γ i and set G i = Stab G (x i ). Let F={G i : i I}. We now define a geometry Γ(G, F) where the objects of type i in Γ(G, F) are the right cosets of G i in G and for G i x and G j y (x, y G, i, j I) G i x G j y whenever G i x G j y. Also by letting G act upon Γ(G, F) by right multiplication we see that Γ(G, F) is a flag transitive geometry for G. Moreover Γ and Γ(G, F) are isomorphic geometries for G. So we shall be studying geometries of the form Γ(G, F), where G = M 22 and G i is a maximal subgroup of G for all i I. A numerical summary of our results is given in 1 Theorem. Suppose G = M 22 and Γ is a residually connected flag transitive geometry for G with Stab G (x) a maximal subgroup of G for all x Γ. (i) If Γ has rank 2, then, up to conjugacy in AutG, there are 86 possibilities for Γ. (ii) If Γ has rank 3, then, up to conjugacy in AutG, there are 1239 possibilities for Γ. In Section 2 and 3 we give explicit descriptions of these geometries making heavy use of the degre2 permutation representation for M 22. The descriptions given readily allow further investigation of these geometries.for the verification of these lists and more details we refer the reader to [6]. Beukenhout, in [1], sought to give a wider view of geometries so as to encompass configurations observed in the finite sporadic simple groups. An outgrowth of this has been attemps to catalogue various subcollections of geometries for the finite sporadic simple groups (and other related groups). So - called minimal parabolic geometries and maximal 2-local geometries were investigated in [13] and [14] while geometries satisfying certain additional conditions for a number of (relatively) small order simple groups have been exhaustively examined. See, for example, [2], [5], [7], [8], [9], [10], [11], [12], [15]; the results in most of these papers were obtained using various computer algebra systems. The lists here, by contrast, has been obtained by hand. For the remainder of this paper G will denote M 22, the Mathieu Group of degre2. Also Ω will denote a 24 element set possessing the Steiner system S(24, 8, 5) as described by Curtis s MOG [4]. We will follow the notation of [4]. So Ω = O 1 O 2 O 3 = , where O 1, O 2 and O 3 are the heavy bricks of the MOG. For our later concrete

3 On Rank 2 and Rank 3 Residually Connected Geometries for M descriptions we shall identify G with Stab M24 { } Stab M24 {14}. Here M 24 is the Mathieu group of degre4 which leaves invariant the Steiner system S(24, 8, 5) on Ω. Set Λ = Ω\{, 14}. We shall discuss certain (well-known) combinatorial objects associated with Ω and Λ. An octad of Ω is just an 8-element block of the Steiner system and a subset of Ω is called a dodecad if it is the symmetric difference of two octads of Ω which intersect in a set of size two. The following sets will appear frequently when we describe geometries for G. (i) H={X Λ X {, 14} is an octad of Ω} (hexads of Λ). (ii) H p ={X Λ X {14} is an octad of Ω} (heptads of Λ). (iii) H p ={X Λ X { } is an octad of Ω} (heptads of Λ). (iv) O={X Λ X is an octad of Ω} (octads of Λ). (v) D={X Λ X = 2} (duads of Λ). (vi) D o ={X Λ X is a dodecad of Ω} (dodecads of Λ). (vii) E={X Λ one of X { } and X {14} is a dodecad of Ω} (endecads of Λ). From the [3], the conjugacy classes of the maximal subgroups of G are as follows: Order Index M i Description M 1 = L3 (4) M 1 = Stab G {a}, a Λ M 2 = 2 4 : A 6 M 2 = Stab G {X}, X H M 3 = A7 M 3 = Stab G {X}, X H p M 4 = A7 M 4 = Stab G {X}, X H p M 5 = 2 3 : L 3 (2) M 5 = Stab G {X}, X O M 6 = 2 4 : S 5 M 6 = Stab G {X}, X D M 7 = M10 M 7 = Stab G {X}, X D o M 8 = L2 (11) M 8 = Stab G {X}, X E For i {1,..., 8}, we let M i denote the conjugacy class of M i, M i as given in the previous table. We also set M= 8 i=1 M i; so M consist of all maximal subgroups of G. Also put X= Λ H H p H p O D D o E. For X, Y X we have that X = Y if and only if Stab G {X} = Stab G {Y } except in the case when X, Y E. In this latter case when Y = Λ\X we have Stab G {X} = Stab G {Y }( = L 2 (11)). In order to have a (1 1) correspondence between subgroups in M and appropriate sets of Λ in X, for endecads we shall always choose X E so as X { } is a dodecad of Ω. Suppose G 1 and G 2 are maximal subgroups of G with G 1 G 2. Set G 12 = G 1 G 2. We use M ij (t) to describe {G 1, G 2, G 1 G 2 } according to the following scheme: G 1 M i, G 2 M j (and so G 1 = Stab G (X 1 ) and G 2 = Stab G (X 2 ) for some appropriate subsets X 1 and X 2 of Λ in X) with X 1 X 2 = t. When listing up the rank 2 geometries of G in Theorem 2 the notation M ij (t) frequently suffices to describe the geometries up to conjugacy in AutG. To distinguish

4 110 N. Kilic, P. Rowley those geometries where this is not the case we shall write M ij (t l ) with l {1, 2}; see after Theorem 2 for the details of these cases. Now suppose we have three distinct maximal subgroup of G, G 1, G 2 and G 3. We shall use G 12, G 13, G 23 and G 123 to denote, respectively G 1 G 2, G 1 G 3, G 2 G 3 and G 1 G 2 G 3. We extend the above notation using M ijk (t ij, t ik, t jk ) to indicate that G 1 M i, G 2 M j, G 3 M k with X i X j = t ij, X i X k = t ik and X j X k = t jk. (Here G 1 = Stab G (X i ), G 2 = Stab G (X j ) and G 3 = Stab G (X k ) for suitable subsets X i, X j and X k of Λ in X.) Again we run into the possibilty that in some instances, we need to further subdivide these cases, and we do this using the ad hoc notation M ijk (t ij, t ik, t jk ; l) where l {1, 2, 3}. See the end of Section 2 for further explanation of this notation. There is a further complexity caused by the division in the rank 2 cases mentioned earlier. In order to accommodate this we use, for example M 155 (0, 0, 4 1 ; 2) in Theorem 3 to tell us that, not only is X j X k = 4 but we are in Case 1 of this situation as described in Theorem 2. We note that if two or more of i, j and k are equal, apparently different parameters t ij, t ik, t jk may describe the same situation. For example M 333 (1, 1, 3) and M 333 (3, 1, 1) describe the same configuration as do M 335 (1, 2, 4) and M 335 (1, 4, 2). We remark that the geometry Γ = Γ(G, F) where F = {G 1, G 2, G 3 } is residually connected if and only if G 1 =< G 12, G 13 >, G 2 =< G 12, G 23 > and G 3 =< G 13, G 23 >. Also we observe that the conjugacy classes M 3 and M 4 fuse in AutG. Below we give certain subsets of Λ which will be encountered frequently in our lits. h 1 =, =, h 3 = h 4 =, h 5 =, h 6 = h 1 =, =, h 3 =

5 On Rank 2 and Rank 3 Residually Connected Geometries for M h 4 =, h 5 =, h 6 = h 7 =, h 8 =, h 9 = h 10 = O 4 =, O 5 =, O 6 = O 7 = =, d 2 =, d 3 = e 1 =, =, e 3 = e 3 = Our notation is as in the [3] with the following addition: SD n is the semidihedral group of order n, F n a Frobenious group of order n and (S n S m ) + is the group of even permutation in the permutation group S n S m.

6 112 N. Kilic, P. Rowley 1 Rank 2 geometries of M 22 2 Theorem. Up to conjugacy in AutG there are 86 rank 2 geometries Γ = Γ(G, {G 1, G 2 }) with G 1, G 2 M. These together with the shape of G 12, are listed in the following table. Γ G 12 Γ G 12 Γ G 12 M 11 (0) 2 4 A 5 M 12 (0) A 6 M 12 (1) 2 4 A 5 M 13 (0) L 3 (2) M 13 (1) A 6 M 15 (0) 2 4 S 3 M 15 (1) L 3 (2) M 16 (0) 2 4 S 3 M 16 (1) 2 4 A 5 M 17 (0) M 9 M 17 (1) A 5 M 18 (0) A 5 M 18 (1) A 5 M 22 (0) A 6 M 22 (2) 2 2 S 4 M 23 (1) A 5 M 23 (3) (S 3 S 4 ) + M 25 (0) 2 3 S 4 M 25 (2) S 4 M 25 (4) 2 2 S 4 M 26 (0) 2 S 4 M 26 (1) A 5 M 26 (2) 2 4 S 4 M 27 (2) S 4 M 27 (4) SD 16 M 27 (6) A 6 M 28 (1) A 5 M 28 (3) D 12 M 28 (5) A 5 M 33 (1) M 33 (3) S 4 M 34 (0) L 3 (2) M 34 (2) F 20 M 34 (4) (S 3 S 4 ) + M 35 (0) L 3 (2) M 35 (2) D 12 M 35 (4) S 4 M 36 (0) S 4 M 36 (1) S 4 M 36 (2) S 5 M 37 (2) F 20 M 37 (4) S 3 M 37 (6) M 38 (2) D 12 M 38 (4) S 3 M 38 (6) A 5 M 55 (0) 2 3 S 4 M 55 (2) D 8 M 55 (4 1 ) A 4 M 55 (4 2 ) M 56 (0 1 ) 2 4 (2 S 3 ) M 56 (0 2 ) 2 D 8 M 56 (1) A 4 M 56 (2) 2 2 (2 S 3 ) M 57 (2) S 4 M 57 (4 1 ) S 4 M 57 (4 2 ) 4 M 57 (6) D 8 M 58 (2) D 12 M 58 (4 1 ) 2 2 M 58 (4 2 ) A 4 M 58 (6) D 12 M 66 (0 1 ) D 12 M 66 (0 2 ) 2 3 D 8 M 66 (1) M 67 (0) SD 16 M 67 (1) S 3 M 67 (2 1 ) F 20 M 67 (2 2 ) S 4 M 68 (0) D 12 M 68 (1 1 ) D 10 M 68 (1 2 ) A 4 M 68 (2) D 12 M 77 (4) SD 16 M 77 (6) 2 M 77 (8 1 ) 4 M 77 (8 2 ) S 4 M 78 (4) S 3 M 78 (6 1 ) D 10 M 78 (6 2 ) 2 M 78 (8) S 3 M 88 (3) D 12 M 88 (5 1 ) 2 M 88 (5 2 ) D 10 M 88 (7 1 ) 2 2 M 88 (7 2 ) A 4

7 On Rank 2 and Rank 3 Residually Connected Geometries for M We now define the subgroups for M ij (t l ) (l {1, 2}) by giving the X i X such that G i = Stab G X i (i = 1, 2).

8 114 N. Kilic, P. Rowley M ij (t l ) X 1 X 2 M 55 (4 1 ) O 2 O 6 M 55 (4 2 ) O 2 O 7 M 56 (O 1 ) O 2 {0, 8} M 56 (O 2 ) O 2 {19, 22} M 57 (4 1 ) O 3 M 57 (4 2 ) M 58 (4 1 ) O 7 M 58 (4 2 ) O 3 e 1 M 66 (0 1 ) {11, 17} {19, 22} M 66 (0 2 ) {0, 8} {3, 20} M 67 (2 1 ) {8, 17} M 67 (2 2 ) {11, 17} M 68 (1 1 ) {1, 22} M 68 (1 2 ) {11, 17} e 1 M 77 (8 1 ) M 77 (8 2 ) d 2 M 78 (6 1 ) M 78 (6 2 ) e 1 M 88 (5 1 ) M 88 (5 2 ) M 88 (7 1 ) e 4 M 88 (7 2 ) e 1

9 On Rank 2 and Rank 3 Residually Connected Geometries for M Rank 3 geometries of M 22 3 Theorem. Up to conjugacy in AutG there are 1239 rank 3 residualy connected geometries Γ = Γ(G, {G 1, G 2, G 3 }) with G 1, G 2, G 3 M. These together with the shape of G 123 are listed in the following table below

10 116 N. Kilic, P. Rowley Γ G 123 Γ G 123 M 111 (0, 0, 0) M 112 (0, 0, 0) S 4 M 112 (0, 0, 1) A 5 M 112 (0, 1, 1) 2 4 A 4 M 113 (0, 0, 0) A 4 M 113 (0, 0, 1) S 4 M 113 (0, 1, 1) A 5 M 115 (0, 0, 0) 2 3 M 115 (0, 0, 1) A 4 M 115 (0, 1, 1) S 4 M 116 (0, 0, 0 : 1) M 116 (0, 0, 0 : 2) S 3 M 116 (0, 1, 0) M 117 (0, 1, 0) S 3 M 117 (0, 0, 0) Q 8 M 118 (0, 1, 1) S 3 M 118 (0, 0, 1 : 1) A 4 M 118 (0, 0, 1 : 2) D 10 M 122 (0, 0, 0) M 122 (0, 1, 0) A 5 M 122 (0, 0, 2) D 8 M 122 (1, 0, 2) S 4 M 122 (1, 1, 2) 2 2 A 4 M 123 (0, 0, 1) S 3 M 123 (0, 1, 1) D 10 M 123 (1, 0, 1) A 4 M 123 (0, 0, 3) S 3 M 123 (0, 1, 3) M 123 (1, 0, 3) S 4 M 123 (1, 1, 3) S 4 M 125 (0, 0, 0) S 4 M 125 (0, 1, 0) S 4 M 125 (1, 0, 0) 2 2 D 8 M 125 (0, 1, 2) 4 M 125 (0, 0, 2 : 1) S 3 M 125 (0, 0, 2 : 2) 2 2 M 125 (1, 0, 2) S 3 M 125 (1, 1, 2) A 4 M 125 (0, 0, 4) D 8 M 125 (0, 1, 4) S 4 M 125 (1, 1, 4) S 4 M 125 (1, 0, 4) 2 2 A 4 M 126 (0, 0, 0 : 1) S 3 M 126 (0, 0, 0 : 2) D 8 M 126 (1, 0, 0) 2 3 M 126 (0, 1, 0) S 4 M 126 (0, 0, 1) 2 2 M 126 (1, 0, 1) A 4 M 126 (0, 0, 2) S 4 M 126 (1, 1, 2) 2 4 A 4 M 127 (0, 0, 2) 4 M 127 (1, 0, 2) S 3 M 127 (0, 0, 4) 2 M 127 (0, 1, 4) 2 M 127 (1, 1, 4) 2 2 M 127 (1, 0, 4) Q 8 M 127 (0, 0, 6) M 128 (0, 1, 1) S 3 M 128 (0, 0, 1) D 10 M 128 (1, 0, 1) A 4 M 128 (0, 1, 3 : 1) S 3 M 128 (0, 1, 3 : 2) 2 M 128 (0, 0, 3 : 1) S 3 M 128 (0, 0, 3 : 2) 2 M 128 (1, 0, 3) 2 2

11 On Rank 2 and Rank 3 Residually Connected Geometries for M M 128 (1, 1, 3) 2 2 M 133 (0, 0, 1) 4 M 133 (1, 0, 1) S 3 M 133 (0, 0, 3 : 1) S 3 M 133 (0, 0, 3 : 2) D 8 M 133 (0, 3, 1) S 3 M 133 (1, 1, 3) D 8 M 134 (0, 0, 0) F 21 M 134 (1, 0, 0) S 4 M 134 (0, 0, 2) 2 M 134 (0, 1, 2) 4 M 134 (1, 1, 2) D 10 M 134 (0, 0, 4) S 3 M 134 (1, 1, 4) M 134 (1, 0, 4) S 4 M 135 (0, 1, 0) F 21 M 135 (0, 0, 0) S 4 M 135 (1, 0, 0) S 4 M 135 (0, 1, 2) 2 M 135 (0, 0, 2 : 1) 2 M 135 (0, 0, 2 : 2) 2 2 M 135 (1, 0, 2 : 1) 2 2 M 135 (1, 0, 2 : 2) S 3 M 135 (1, 1, 2) S 3 M 135 (0, 0, 4 : 1) D 8 M 135 (0, 0, 4 : 2) 3 M 135 (1, 1, 4) S 3 M 135 (0, 1, 4) S 3 M 135 (1, 0, 4) D 8 M 136 (0, 0, 0) 2 M 136 (1, 0, 0) S 3 M 136 (0, 1, 1) A 4 M 136 (0, 0, 1 : 1) 3 M 136 (0, 0, 1 : 2) 2 2 M 136 (1, 0, 1) 2 2 M 136 (0, 0, 2) D 8 M 136 (1, 0, 2) S 4 M 136 (1, 1, 2) A 5 M 137 (0, 1, 2) 2 M 137 (0, 0, 2) 4 M 137 (1, 0, 2) 4 M 137 (1, 1, 2) D 10 M 137 (0, 0, 4) 1 M 137 (1, 0, 4) 2 M 137 (0, 0, 6) 4 M 137 (0, 1, 6) S 3 M 137 (1, 1, 6) S 3 M 138 (0, 1, 2 : 1) 2 M 138 (0, 1, 2 : 2) 2 2 M 138 (0, 0, 2) 2 M 138 (1, 0, 2 : 1) 2 2 M 138 (1, 0, 2 : 2) S 3 M 138 (1, 1, 2) S 3 M 138 (0, 1, 4) 1 M 138 (0, 0, 4 : 1) 2 M 138 (0, 0, 4 : 2) 3 M 138 (1, 0, 4) 2 M 138 (0, 0, 6) S 3 M 138 (1, 1, 6) D 10 M 138 (0, 1, 6) A 4 M 155 (0, 1, 0) S 4 M 155 (0, 0, 2) 2 M 155 (0, 1, 2) 2 M 155 (1, 1, 2) 4 M 155 (0, 0, 4 1 : 1) 3 M 155 (0, 0, 4 1 : 2) 2 2 M 155 (1, 0, 4 1 ) 3 M 155 (1, 1, 4 1 ) 3 M 155 (0, 0, 4 2 ) 2 2 M 155 (0, 1, 4 2 ) D 8

12 118 N. Kilic, P. Rowley M 155 (1, 1, 4 2 ) D 8 M 156 (0, 0, 0 1 ) M 156 (1, 0, 0 1 ) S 4 M 156 (0, 0, 0 2 ) 2 M 156 (1, 0, 0 2 ) 2 M 156 (0, 1, 0 2 ) 2 3 M 156 (0, 0, 1) 1 M 156 (1, 0, 1) 3 M 156 (0, 0, 2 : 1) S 3 M 156 (0, 0, 2 : 2) 2 3 M 156 (1, 0, 2) 2 3 M 156 (1, 1, 2) S 4 M 157 (1, 0, 2) 4 M 157 (0, 0, 2) S 3 M 157 (0, 0, 4 1 ) 4 M 157 (1, 0, 4 1 ) S 3 M 157 (0, 0, 4 2 ) 1 M 157 (1, 0, 4 2 ) 1 M 157 (1, 1, 4 2 ) 1 M 157 (0, 1, 4 2 : 1) S 3 M 157 (0, 1, 4 2 : 2) 2 M 157 (0, 0, 6) 2 M 157 (1, 0, 6) 4 M 158 (0, 1, 2 : 1) 2 M 158 (0, 1, 2 : 2) 2 2 M 158 (1, 0, 2) 2 M 158 (0, 0, 2) 2 2 M 158 (1, 1, 2) S 3 M 158 (0, 1, 4 1 : 1) 2 M 158 (0, 1, 4 1 : 2) 1 M 158 (0, 0, 4 1 : 1) 1 M 158 (0, 0, 4 1 : 2) 2 M 158 (1, 1, 4 1 ) 2 M 158 (1, 0, 4 1 ) 2 M 158 (0, 1, 4 2 ) 2 M 158 (0, 0, 4 2 ) 2 M 158 (1, 1, 4 2 ) 3 M 158 (1, 0, 4 2 ) 3 M 158 (1, 1, 6) 2 M 158 (0, 0, 6 : 1) 2 2 M 158 (0, 0, 6 : 2) 2 M 158 (0, 1, 6 : 1) S 3 M 158 (0, 1, 6 : 2) 2 2 M 158 (1, 0, 6) S 3 M 166 (0, 0, 0 1 : 1) 2 2 M 166 (0, 0, 0 1 : 2) 2 M 166 (1, 0, 0 1 ) S 3 M 166 (0, 0, 0 2 ) 2 2 M 166 (0, 0, 1) 3 M 167 (0, 1, 0) 2 M 167 (0, 0, 0) 2 M 167 (1, 0, 0) Q 8 M 167 (0, 0, 1 : 1) 2 M 167 (0, 0, 1 : 2) 1 M 167 (0, 1, 2 1 ) 2 M 167 (0, 0, 2 1 ) 4 M 167 (1, 1, 2 1 ) D 10 M 167 (0, 0, 2 2 ) 4 M 168 (0, 0, 1 1 ) 2 M 168 (0, 1, 1 1 ) 2 M 168 (0, 0, 1 2 ) 2 M 168 (0, 1, 1 2 ) 2 M 168 (0, 0, 0 : 1) 2 M 168 (0, 0, 0 : 2) 2 2 M 168 (0, 1, 0 : 1) 2 M 168 (0, 1, 0 : 2) 2 2 M 168 (0, 1, 0 : 3) S 3 M 168 (1, 0, 0) S 3

13 On Rank 2 and Rank 3 Residually Connected Geometries for M M 168 (0, 0, 2 : 1) 2 M 168 (0, 0, 2 : 2) 2 2 M 168 (0, 1, 2 : 1) 2 M 168 (0, 1, 2 : 2) 2 2 M 168 (1, 0, 2) S 3 M 177 (0, 1, 4) 2 M 177 (1, 1, 4) 2 2 M 177 (0, 0, 4) Q 8 M 177 (0, 0, 6) 1 M 177 (0, 0, 8 1 ) 1 M 177 (1, 0, 8 1 ) 1 M 177 (1, 1, 8 1 : 1) 1 M 177 (1, 1, 8 1 : 2) 2 M 177 (1, 1, 8 2 ) 4 M 178 (0, 0, 6 1 ) 2 M 178 (0, 1, 6 1 ) 2 M 178 (0, 0, 6 2 ) 1 M 178 (0, 1, 6 2 ) 1 M 178 (0, 0, 8) 1 M 178 (0, 1, 8) 2 M 178 (0, 1, 4) 1 M 178 (0, 0, 4) 2 M 188 (1, 0, 3) 2 M 188 (0, 0, 3) 2 2 M 188 (1, 1, 3) 2 2 M 188 (0, 0, 5 1 ) 1 M 188 (1, 0, 5 1 ) 1 M 188 (0, 1, 5 1 ) 1 M 188 (1, 1, 5 1 ) 1 M 188 (0, 0, 5 2 ) 2 M 188 (1, 0, 5 2 ) 2 M 188 (1, 1, 5 2 ) 2 M 188 (0, 0, 7 1 : 1) 1 M 188 (0, 0, 7 1 : 2) 2 M 188 (1, 1, 7 1 : 1) 1 M 188 (1, 1, 7 1 : 2) 2 M 188 (1, 0, 7 1 ) 2 M 188 (1, 1, 7 2 ) 2 M 188 (0, 0, 7 2 ) 2 M 188 (1, 0, 7 2 ) 3 M 222 (0, 2, 2) D 8 M 222 (0, 0, 2) S 4 M 222 (2, 2, 2 : 1) 3 M 222 (2, 2, 2 : 2) M 223 (0, 3, 1) S 3 M 223 (0, 1, 1) D 10 M 223 (0, 3, 3) M 223 (2, 3, 1 : 1) 2 M 223 (2, 3, 1 : 2) S 3 M 223 (2, 3, 3 : 1) S 3 M 223 (2, 3, 3 : 2) 2 2 M 223 (2, 1, 1) 2 2 M 225 (0, 2, 2) 2 M 225 (0, 4, 2) D 8 M 225 (0, 0, 4) S 4 M 225 (2, 2, 2 : 1) 1 M 225 (2, 2, 2 : 2) 2 M 225 (2, 2, 4) 3 M 225 (2, 2, 0) 2 2 M 225 (2, 4, 4) 2 3 M 225 (2, 0, 0) M 226 (0, 0, 1) S 3 M 226 (0, 0, 0) D 8 M 226 (0, 0, 2) S 4 M 226 (2, 1, 0) 2 M 226 (2, 0, 0 : 1) 2 M 226 (2, 0, 0 : 2) 2 3 M 226 (2, 1, 2) A 4 M 226 (2, 0, 2) 2 D 8 M 227 (2, 4, 4 : 1) 2 2 M 227 (2, 4, 4 : 2) 1 M 227 (2, 4, 2) 1 M 227 (2, 6, 4) D 8

14 120 N. Kilic, P. Rowley M 227 (0, 4, 2) 2 M 227 (0, 4, 4) 2 M 228 (0, 3, 3) 2 M 228 (0, 3, 5) S 3 M 228 (0, 3, 1) S 3 M 228 (0, 1, 5) D 10 M 228 (2, 3, 3 : 1) 1 M 228 (2, 3, 3 : 2) 2 M 228 (2, 3, 3 : 3) 2 2 M 228 (2, 3, 5 : 1) 2 M 228 (2, 3, 5 : 2) 2 2 M 228 (2, 1, 3 : 1) 2 M 228 (2, 1, 3 : 2) 2 2 M 228 (2, 5, 5) S 3 M 228 (2, 1, 1) S 3 M 228 (2, 5, 1) A 4 M 233 (3, 1, 1) 2 M 233 (3, 3, 3 : 1) 2 M 233 (3, 3, 3 : 2) 2 2 M 233 (3, 3, 1) 4 M 233 (3, 1, 3) S 3 M 233 (1, 1, 1) 2 M 233 (1, 1, 3 : 1) 3 M 233 (1, 1, 3 : 2) 2 M 234 (3, 1, 2) 2 M 234 (3, 3, 2 : 1) 2 M 234 (3, 3, 2 : 2) 4 M 234 (3, 3, 4) 2 2 M 234 (3, 1, 0) S 3 M 234 (3, 1, 4) S 3 M 234 (1, 1, 2) 1 M 234 (1, 1, 4) 2 2 M 235 (3, 2, 2 : 1) 1 M 235 (3, 2, 2 : 2) 2 M 235 (3, 2, 4) 2 M 235 (3, 4, 4 : 1) 2 2 M 235 (3, 4, 4 : 2) S 3 M 235 (3, 4, 2) 2 2 M 235 (3, 0, 2) 2 2 M 235 (3, 2, 0) S 3 M 235 (3, 0, 0) S 4 M 235 (1, 2, 2) 1 M 235 (1, 4, 2 : 1) 1 M 235 (1, 4, 2 : 2) 2 2 M 235 (1, 2, 4) 2 M 235 (1, 0, 4) 2 2 M 235 (1, 0, 2) 2 2 M 235 (1, 2, 0) S 3 M 235 (1, 4, 4) S 3 M 235 (1, 4, 0) A 4 M 236 (3, 0, 1) 2 M 236 (3, 0, 0 : 1) 2 M 236 (3, 0, 0 : 2) 2 2 M 236 (3, 0, 0 : 3) S 3 M 236 (3, 1, 2) S 3 M 236 (3, 2, 1) D 8 M 236 (3, 0, 2) D 12 M 236 (3, 2, 2) S 4 M 236 (1, 0, 1) 2 M 236 (1, 0, 0 : 1) 2 M 236 (1, 0, 0 : 2) 2 2 M 236 (1, 1, 0 : 1) 2 M 236 (1, 1, 0 : 2) 3 M 236 (1, 0, 2) 2 2 M 237 (3, 4, 4 : 1) 1 M 237 (3, 4, 4 : 2) 2 M 237 (3, 4, 2 : 1) 2 M 237 (3, 4, 2 : 2) 4 M 237 (3, 2, 2) 2 M 237 (3, 6, 6) M 237 (1, 4, 2 : 1) 1 M 237 (1, 4, 2 : 2) 2 M 237 (1, 4, 4 : 1) 1 M 237 (1, 4, 4 : 2) 2 M 237 (1, 4, 6) 2 M 237 (1, 2, 6) 2 M 237 (1, 2, 2) 2 M 237 (1, 6, 2) D 10

15 On Rank 2 and Rank 3 Residually Connected Geometries for M M 238 (3, 3, 4 : 1) 1 M 238 (3, 3, 4 : 2) 2 M 238 (3, 3, 2 : 1) 1 M 238 (3, 3, 2 : 2) 2 M 238 (3, 3, 2 : 3) 2 2 M 238 (3, 1, 2 : 1) 2 M 238 (3, 1, 2 : 2) S 3 M 238 (3, 1, 4) 2 M 238 (3, 3, 6) 2 2 M 238 (3, 5, 6) S 3 M 238 (1, 1, 4) 1 M 238 (1, 3, 2 : 1) 1 M 238 (1, 3, 2 : 2) 2 M 238 (1, 3, 4 : 1) 1 M 238 (1, 3, 4 : 2) 2 M 238 (1, 5, 2) 2 M 238 (1, 3, 6) 2 M 238 (1, 5, 4) 2 M 238 (1, 1, 2) 2 2 M 255 (2, 2, 4 1 : 1) 1 M 255 (2, 2, 4 1 : 2) 2 M 255 (2, 2, 2) 1 M 255 (0, 2, 4 1 ) 2 M 255 (2, 4, 4 1 ) 2 M 255 (0, 2, 2) 2 M 255 (4, 4, 4 1 ) 3 M 255 (2, 2, 0) 2 2 M 255 (2, 2, 4 2 ) 2 2 M 255 (2, 4, 4 2 ) 2 2 M 255 (4, 0, 2) 2 2 M 255 (4, 4, 4 2 ) 2 3 M 255 (0, 0, 4 2 ) 2 4 M 255 (4, 0, 0) M 256 (0, 0, 1) 3 M 256 (0, 1, 0 2 ) 2 2 M 256 (0, 0, 2) 2 3 M 256 (0, 0, 0 2 ) 2 3 M 256 (0, 2, 0 1 ) M 256 (2, 0, 1 : 1) 1 M 256 (2, 0, 1 : 2) 2 M 256 (2, 1, 0 2 ) 2 M 256 (2, 1, 2) 2 M 256 (2, 0, 2) 2 M 256 (2, 2, 1) 3 M 256 (2, 2, 0 2 ) 2 2 M 256 (2, 1, 0 1 ) S 3 M 256 (2, 0, 0 1 ) D 8 M 256 (4, 0, 1) 2 M 256 (4, 1, 0 1 ) 2 2 M 256 (4, 1, 2) S 3 M 256 (4, 2, 2) M 257 (0, 4, 4 2 ) 1 M 257 (0, 2, 4 2 ) 2 M 257 (0, 4, 6) 2 2 M 257 (0, 4, 2) 2 2 M 257 (2, 4, 2) 1 M 257 (2, 4, 4 2 ) 1 M 257 (2, 4, 6 : 1) 1 M 257 (2, 4, 6 : 2) 2 M 257 (2, 4, 4 1 : 1) 1 M 257 (2, 4, 4 1 : 2) 2 M 257 (2, 2, 4 2 : 1) 1 M 257 (2, 2, 4 2 : 2) 2 M 257 (2, 6, 4 2 ) 2 M 257 (4, 2, 4 2 ) 1 M 257 (4, 4, 4 2 ) 1 M 257 (4, 4, 6) 1 M 257 (4, 4, 4 1 ) 3 M 257 (4, 4, 2) D 8 M 258 (0, 3, 4 2 : 1) 3 M 258 (0, 3, 4 2 : 2) 2 2 M 258 (0, 1, 6) 2 2 M 258 (0, 5, 2) 2 2 M 258 (0, 3, 2) 2 2 M 258 (0, 3, 6) 2 2 M 258 (2, 3, 2 : 1) 1 M 258 (2, 3, 2 : 2) 2

16 122 N. Kilic, P. Rowley M 258 (2, 3, 4 1 ) 1 M 258 (2, 5, 4 1 ) 1 M 258 (2, 1, 4 1 ) 1 M 258 (2, 3, 4 2 : 1) 1 M 258 (2, 3, 4 2 : 2) 2 M 258 (2, 3, 6 : 1) 1 M 258 (2, 3, 6 : 2) 2 M 258 (2, 1, 4 2 : 1) 2 M 258 (2, 1, 4 2 : 2) 2 M 258 (2, 5, 4 2 : 1) 2 M 258 (2, 5, 4 2 : 2) 3 M 258 (2, 5, 2) 2 M 258 (2, 1, 6) 2 M 258 (2, 5, 6 : 1) 2 M 258 (2, 5, 6 : 2) 2 M 258 (2, 1, 2) 2 M 258 (4, 3, 4 1 : 1) 1 M 258 (4, 3, 4 1 : 2) 2 M 258 (4, 3, 4 2 ) 1 M 258 (4, 1, 4 1 ) 2 M 258 (4, 5, 4 1 ) 2 M 258 (4, 3, 6 : 1) 2 M 258 (4, 3, 6 : 2) 2 2 M 258 (4, 3, 2 : 1) 2 M 258 (4, 3, 2 : 2) 2 2 M 258 (4, 5, 6) 2 2 M 258 (4, 1, 2) S 3 M 266 (1, 1, 0 1 ) 1 M 266 (0, 0, 0 1 ) 1 M 266 (0, 1, 0 2 ) 2 M 266 (0, 0, 1) 3 M 266 (1, 1, 0 2 ) 2 2 M 266 (2, 0, 0 1 ) 2 2 M 266 (1, 0, 0 2 ) 2 2 M 266 (0, 0, 1) 2 2 M 266 (1, 2, 0 1 ) 2 2 M 266 (2, 0, 0 2 ) 2 D 8 M 266 (2, 0, 0 1 ) 2 2 M 267 (0, 4, 0 : 1) 1 M 267 (0, 4, 0 : 2) 2 M 267 (0, 4, 1) 1 M 267 (0, 4, 2 1 ) 1 M 267 (0, 2, 2 1 ) 2 M 267 (0, 2, 0 : 1) 2 M 267 (0, 2, 0 : 2) D 8 M 267 (0, 6, 0) D 8 M 267 (0, 4, 2 2 ) 2 M 267 (1, 2, 0) 1 M 267 (1, 4, 2 2 ) 1 M 267 (1, 4, 0) 1 M 267 (1, 4, 2 1 ) 2 M 267 (1, 2, 2 1 ) 2 M 267 (1, 6, 2 1 ) D 10 M 267 (2, 4, 1) 2 M 267 (2, 2, 0) 2 2 M 267 (2, 4, 2 1 ) 4 M 267 (2, 4, 2 2 ) D 8 M 268 (0, 3, 2) 1 M 268 (0, 3, 0) 1 M 268 (0, 5, 1 1 ) 2 M 268 (0, 1, 1 1 ) 2 M 268 (0, 1, 2) 2 M 268 (0, 5, 0 : 1) 2 M 268 (0, 5, 0 : 2) 2 2 M 268 (0, 3, 1 1 ) 2 M 268 (0, 5, 2) 2 2 M 268 (0, 1, 0) 2 2 M 268 (1, 3, 0 : 1) 1 M 268 (1, 3, 0 : 2) 2 M 268 (1, 3, 2 : 1) 1 M 268 (1, 3, 2 : 2) 2 M 268 (1, 5, 2) 2 M 268 (1, 1, 0) 2 M 268 (2, 3, 1 1 ) 2 M 268 (2, 3, 0) 2 2

17 On Rank 2 and Rank 3 Residually Connected Geometries for M M 268 (2, 3, 2) 2 2 M 268 (2, 3, 1 2 ) 2 2 M 268 (2, 5, 2) S 3 M 268 (2, 1, 0) S 3 M 277 (4, 4, 8 2 ) 1 M 277 (2, 2, 8 1 : 1) 1 M 277 (2, 2, 8 1 : 2) 2 M 277 (4, 2, 8 1 : 1) 1 M 277 (4, 2, 8 1 : 2) 2 M 277 (4, 4, 8 1 ) 1 M 277 (2, 4, 4) 1 M 277 (4, 4, 6) 1 M 277 (4, 4, 4) 1 M 277 (4, 6, 8 1 ) 2 M 277 (2, 2, 8 1 ) 2 M 277 (2, 2, 4) 2 2 M 277 (2, 6, 4) D 8 M 278 (4, 1, 4 : 1) 1 M 278 (4, 1, 4 : 2) 2 M 278 (4, 3, 4) 1 M 278 (4, 1, 6 2 ) 1 M 278 (4, 3, 6 2 ) 1 M 278 (4, 3, 6 1 ) 1 M 278 (4, 3, 8 : 1) 1 M 278 (4, 3, 8 : 2) 2 M 278 (4, 5, 6 2 ) 1 M 278 (4, 5, 8 : 1) 1 M 278 (4, 5, 8 : 2) 2 M 278 (4, 1, 8) 2 M 278 (4, 1, 6 1 ) 2 M 278 (4, 5, 4) 2 M 278 (4, 5, 6 1 ) 2 M 278 (4, 3, 4) 2 M 288 (3, 5, 5 1 ) 1 M 288 (3, 3, 5 1 ) 1 M 288 (5, 1, 5 1 ) 1 M 288 (3, 3, 3) 1 M 288 (1, 3, 5 1 ) 1 M 288 (3, 3, 5 2 ) 1 M 288 (3, 3, 7 1 ) 1 M 288 (3, 5, 7 1 : 1) 1 M 288 (3, 5, 7 1 : 2) 2 M 288 (3, 3, 7 2 ) 1 M 288 (3, 5, 7 2 ) 2 M 288 (5, 5, 7 1 ) 2 M 288 (1, 3, 7 1 : 1) 1 M 288 (1, 3, 7 1 : 2) 2 M 288 (1, 3, 5 2 ) 2 M 288 (5, 3, 5 2 ) 2 M 288 (3, 5, 3) 2 M 288 (3, 1, 3) 2 M 288 (1, 3, 7 2 ) 2 M 288 (5, 5, 7 2 ) 3 M 288 (1, 1, 7 2 ) 3 M 288 (1, 5, 3) 2 2 M 333 (1, 1, 3 : 1) 1 M 333 (1, 1, 3 : 2) S 3 M 333 (1, 1, 1) 2 M 333 (3, 1, 3) 2 M 333 (3, 3, 3 : 1) 1 M 333 (3, 3, 3 : 2) 2 M 334 (3, 2, 2 : 1) 1 M 334 (3, 2, 2 : 2) 2 M 334 (1, 2, 2 : 1) 1 M 334 (1, 2, 2 : 2) 2 M 334 (3, 2, 4) 2 M 334 (1, 2, 4 : 1) 2 M 334 (1, 2, 4 : 2) 4 M 334 (3, 2, 0) 2 M 334 (1, 2, 0) 4 M 334 (3, 4, 4 : 1) 2 2 M 334 (3, 4, 4 : 2) S 3 M 334 (1, 4, 0) S 3 M 334 (3, 0, 0) A 4 M 334 (1, 4, 4) 3 2 2

18 124 N. Kilic, P. Rowley M 335 (1, 2, 2) 1 M 335 (1, 2, 4 : 1) 1 M 335 (1, 2, 4, : 2) 2 M 335 (3, 4, 2 : 1) 1 M 335 (3, 4, 2 : 2) 2 M 335 (3, 4, 4 : 1) 1 M 335 (3, 4, 4 : 2) 2 M 335 (3, 0, 2) 2 M 335 (1, 4, 4) 2 M 335 (1, 2, 0) S 3 M 335 (3, 2, 2) 1 M 335 (1, 0, 4) 4 M 336 (3, 0, 1 : 1) 1 M 336 (3, 0, 1 : 2) 2 M 336 (1, 1, 0 : 1) 1 M 336 (1, 1, 0 : 2) 2 M 336 (1, 0, 0) 2 M 336 (3, 0, 0) 1 M 336 (3, 2, 1) 2 M 336 (3, 2, 0) 2 2 M 336 (1, 2, 0 : 1) 4 M 336 (1, 2, 0 : 2) S 3 M 336 (3, 2, 2) D 8 M 337 (3, 4, 2 : 1) 1 M 337 (3, 4, 2 : 2) 2 M 337 (3, 6, 4 : 1) 1 M 337 (3, 6, 4 : 2) 2 M 337 (3, 2, 2 : 1) 1 M 337 (3, 2, 2 : 2) 2 M 337 (1, 4, 2 : 1) 1 M 337 (1, 4, 2 : 2) 2 M 337 (1, 6, 2) 2 M 337 (3, 6, 6) 2 M 337 (1, 6, 4) 1 M 337 (3, 6, 2) 4 M 337 (1, 2, 2) 4 M 338 (3, 2, 4 : 1) 1 M 338 (3, 2, 4 : 2) 2 M 338 (1, 4, 4) 1 M 338 (3, 4, 4) 1 M 338 (3, 2, 2 : 1) 1 M 338 (3, 2, 2 : 2) 2 M 338 (1, 2, 4 : 1) 1 M 338 (1, 2, 4 : 2) 2 M 338 (1, 2, 2) 2 M 338 (1, 6, 4) 2 M 338 (3, 4, 6) 2 M 338 (1, 2, 6) 2 M 338 (3, 6, 2) 2 2 M 338 (3, 6, 6) S 3 M 345 (2, 2, 2) 1 M 345 (2, 2, 4 : 1) 1 M 345 (2, 2, 4 : 2) 2 M 345 (2, 4, 4 : 1) 1 M 345 (2, 4, 4 : 2) 2 M 345 (4, 2, 2) 1 M 345 (4, 4, 4) 2 M 345 (0, 4, 2) 2 M 345 (0, 2, 2) 2 M 345 (4, 2, 4) 2 M 345 (2, 4, 0) 4 M 345 (4, 0, 2) S 3 M 345 (0, 0, 4) A 4 M 345 (4, 0, 0) S 4 M 346 (4, 0, 0) 2 M 346 (4, 1, 2) S 3 M 346 (4, 2, 2) D 12 M 346 (0, 0, 1) 2 M 346 (0, 0, 0) S 3 M 346 (0, 2, 0) D 8 M 346 (2, 0, 0 : 1) 1 M 346 (2, 0, 0 : 2) 2

19 On Rank 2 and Rank 3 Residually Connected Geometries for M M 346 (2, 1, 0 : 1) 1 M 346 (2, 1, 0 : 2) 2 M 346 (2, 2, 1) 2 M 346 (2, 2, 0) 2 M 347 (4, 2, 4) 2 M 347 (4, 6, 4) 2 M 347 (4, 2, 2 : 1) 2 M 347 (4, 2, 2 : 2) 4 M 347 (4, 6, 6) 4 M 347 (0, 2, 4) 2 M 347 (0, 2, 6) 4 M 347 (2, 6, 4 : 1) 1 M 347 (2, 6, 4 : 2) 2 M 347 (2, 2, 4 : 1) 1 M 347 (2, 2, 4 : 2) 2 M 347 (2, 2, 2) 1 M 347 (2, 6, 2) 2 M 347 (2, 6, 6) 4 M 348 (4, 4, 5 : 1) 1 M 348 (4, 4, 5 : 2) 2 M 348 (4, 2, 3 : 1) 1 M 348 (4, 2, 3 : 2) 2 M 348 (4, 4, 3 : 1) 1 M 348 (4, 4, 3 : 2) 2 M 348 (4, 2, 5) 2 2 M 348 (4, 6, 5 : 1) 2 2 M 348 (4, 6, 5 : 2) S 3 M 348 (4, 2, 1 : 1) 2 2 M 348 (4, 2, 1 : 2) S 3 M 348 (0, 4, 3) 1 M 348 (0, 2, 3) 2 M 348 (0, 2, 5 : 1) 2 M 348 (0, 2, 5 : 2) 2 2 M 348 (0, 4, 5) 2 M 348 (0, 6, 1) A 4 M 348 (2, 4, 5 : 1) 1 M 348 (2, 4, 5 : 2) 2 M 348 (2, 4, 3 : 1) 1 M 348 (2, 4, 3 : 2) 2 M 348 (2, 6, 3 : 1) 1 M 348 (2, 6, 3 : 2) 2 M 348 (2, 2, 3 : 1) 1 M 348 (2, 2, 3 : 2) 2 M 348 (2, 4, 1 : 1) 1 M 348 (2, 4, 1 : 2) 2 M 348 (2, 2, 1) 2 M 348 (2, 6, 5) 2 M 355 (4, 4, 4 1 ) 1 M 355 (2, 2, 4 1 : 1) 1 M 355 (2, 2, 4 1 : 2) 2 M 355 (2, 2, 4 2 ) 1 M 355 (2, 4, 4 1 ) 1 M 355 (2, 2, 2 : 1) 1 M 355 (2, 2, 2 : 2) 2 M 355 (2, 4, 2) 1 M 355 (2, 4, 4 2 ) 2 M 355 (0, 2, 2) 2 M 355 (0, 2, 4 1 ) 2 M 355 (0, 4, 2) 2 M 355 (4, 4, 2) 2 M 355 (4, 4, 4 2 ) 2 M 355 (0, 2, 4 2 ) 2 2 M 355 (4, 2, 0) 2 2 M 356 (4, 1, 0 2 ) 1 M 356 (4, 0, 0 2 ) 1 M 356 (4, 0, 1) 1 M 356 (4, 2, 1) 1 M 356 (4, 1, 2) 2 M 356 (4, 0, 2) 2 2 M 356 (4, 2, 2) 2 2 M 356 (4, 0, 0 1 ) S 3 M 356 (4, 2, 0 1 ) D 8 M 356 (2, 0, 1) 1 M 356 (2, 0, 0 2 : 1) 1 M 356 (2, 0, 0 2 : 2) 2

20 126 N. Kilic, P. Rowley M 356 (2, 1, 0 2 ) 1 M 356 (2, 1, 2) 2 M 356 (2, 1, 0 1 : 1) 2 M 356 (2, 1, 0 1 : 2) 2 2 M 356 (2, 0, 2) 2 M 356 (2, 2, 0 2 ) 2 M 356 (2, 2, 1) 2 M 356 (2, 0, 0 1 ) 2 2 M 356 (0, 0, 1) 3 M 356 (0, 0, 2) S 3 M 356 (0, 2, 0 2 ) D 8 M 357 (2, 2, 6 : 1) 1 M 357 (2, 2, 6 : 2) 2 M 357 (2, 6, 4 2 ) 1 M 357 (2, 2, 2) 2 M 357 (2, 2, 4 1 ) 2 M 357 (2, 6, 6) 2 M 357 (2, 6, 2) 2 M 357 (2, 6, 4 1 ) 2 M 357 (4, 2, 4 2 ) 1 M 357 (4, 6, 4 2 ) 1 M 357 (4, 2, 2) 2 M 357 (4, 2, 4 1 ) 2 M 357 (4, 2, 6) 2 M 357 (4, 6, 6) 2 M 357 (0, 4, 4 2 ) 1 M 357 (0, 2, 6) 2 M 357 (0, 6, 2) S 3 M 358 (2, 4, 6) 1 M 358 (2, 4, 2) 1 M 358 (2, 6, 4 1 ) 1 M 358 (2, 2, 4 1 ) 1 M 358 (2, 2, 4 2 ) 1 M 358 (2, 4, 4 1 ) 1 M 358 (2, 4, 4 2 ) 1 M 358 (2, 2, 2) 1 M 358 (2, 2, 6) 1 M 358 (2, 6, 4 2 ) 2 M 358 (2, 6, 2) 2 M 358 (2, 2, 6) 2 M 358 (4, 4, 6) 1 M 358 (4, 2, 4 1 ) 1 M 358 (4, 2, 4 2 : 1) 1 M 358 (4, 2, 4 2 : 2) 2 M 358 (4, 4, 4 2 ) 1 M 358 (4, 2, 2 : 1) 1 M 358 (4, 2, 2 : 2) 2 M 358 (4, 4, 2 : 1) 1 M 358 (4, 4, 2 : 2) 2 M 358 (4, 4, 4 1 ) 1 M 358 (4, 6, 6) 2 M 358 (4, 4, 6) 2 M 358 (4, 2, 6) 2 M 358 (4, 6, 4 2 ) 3 M 358 (0, 4, 4 1 ) 1 M 358 (0, 2, 4 1 ) 2 M 358 (0, 2, 6) 2 M 358 (0, 4, 2) 2 M 358 (0, 2, 4 2 ) 2 2 M 358 (0, 6, 2) S 3 M 366 (1, 1, 0 1 ) 1 M 366 (1, 0, 0 1 : 1) 1 M 366 (1, 0, 0 1 : 2) 2 M 366 (0, 0, 0 1 : 1) 1 M 366 (0, 0, 0 1 : 2) 2 M 366 (0, 0, 1) 1 M 366 (1, 2, 0 1 ) 2 M 366 (2, 0, 0 1 ) 2 M 366 (0, 0, 0 2 ) 2 M 366 (0, 1, 0 2 ) 2 M 366 (2, 1, 0 2 ) D 8 M 366 (2, 2, 1) A 4 M 367 (1, 4, 0) 1 M 367 (1, 2, 0 : 1) 1

21 On Rank 2 and Rank 3 Residually Connected Geometries for M M 367 (1, 2, 0 : 2) 4 M 367 (1, 4, 2 1 : 1) 1 M 367 (1, 4, 2 1 : 2) 2 M 367 (1, 6, 2 2 ) 2 M 367 (1, 2, 2 1 ) 2 M 367 (1, 6, 2 1 ) 2 M 367 (1, 2, 2 2 ) 2 M 367 (0, 4, 0 : 1) 1 M 367 (0, 4, 0 : 2) 1 M 367 (0, 2, 1 : 1) 1 M 367 (0, 2, 1 : 2) 2 M 367 (0, 6, 0) 2 M 367 (0, 2, 0) 2 M 367 (0, 2, 2 1 ) 1 M 367 (0, 6, 1) 1 M 367 (0, 2, 2 2 ) 2 M 367 (0, 6, 2 1 ) 4 M 367 (2, 4, 0) 2 M 367 (2, 4, 2 1 ) 2 M 367 (2, 2, 1) 2 M 367 (2, 2, 0) 2 M 367 (2, 6, 2 1 ) 4 M 367 (2, 6, 2 2 ) S 3 M 368 (0, 4, 2 : 1) 1 M 368 (0, 4, 2 : 2) 2 M 368 (0, 2, 1 1 ) 1 M 368 (0, 4, 1 2 ) 1 M 368 (0, 4, 0 : 1) 1 M 368 (0, 4, 0 : 2) 2 M 368 (0, 2, 1 2 ) 1 M 368 (0, 2, 2 : 1) 1 M 368 (0, 2, 2 : 2) 2 M 368 (0, 2, 2 : 3) 2 2 M 368 (0, 6, 1 1 ) 2 M 368 (0, 2, 0 : 1) 2 M 368 (0, 2, 0 : 2) 2 2 M 368 (0, 6, 0) 2 M 368 (1, 2, 0 : 1) 1 M 368 (1, 2, 0 : 2) 2 M 368 (1, 2, 2) 1 M 368 (1, 4, 0 : 1) 1 M 368 (1, 4, 0 : 2) 2 M 368 (1, 6, 2) 2 M 368 (2, 4, 1 2 ) 1 M 368 (2, 4, 0) 2 M 368 (2, 4, 2) 2 M 368 (2, 2, 0 : 1) 2 M 368 (2, 2, 0 : 2) 2 2 M 368 (2, 2, 1 1 ) 2 M 368 (2, 2, 1 2 ) 3 M 368 (2, 6, 2) 2 2 M 377 (6, 2, 6) 1 M 377 (2, 2, 6) 1 M 377 (2, 2, 8 1 ) 1 M 377 (2, 4, 4 : 1) 1 M 377 (2, 4, 4 : 2) 2 M 377 (2, 2, 8 2 ) 2 M 377 (2, 6, 4) 4 M 377 (4, 6, 8 1 ) 1 M 377 (4, 2, 8 1 ) 1 M 377 (6, 2, 4) 2 M 377 (4, 6, 4) 2 M 377 (6, 6, 8 2 ) S 3 M 377 (6, 6, 8 1 ) 1 M 378 (6, 4, 8 : 1) 1 M 378 (6, 4, 8 : 2) 2 M 378 (6, 2, 6 2 ) 1 M 378 (6, 4, 4) 1 M 378 (6, 2, 4 : 1) 1 M 378 (6, 2, 4 : 2) 2 M 378 (6, 2, 6 1 ) 2 M 378 (6, 6, 8) 2 M 378 (2, 2, 8 : 1) 1 M 378 (2, 2, 8 : 2) 2

22 128 N. Kilic, P. Rowley M 378 (2, 2, 4 : 1) 1 M 378 (2, 2, 4 : 2) 2 M 378 (2, 4, 6 1 ) 1 M 378 (2, 2, 6 2 ) 1 M 378 (2, 4, 8 : 1) 1 M 378 (2, 4, 8 : 2) 2 M 378 (2, 4, 4) 1 M 378 (2, 6, 6 2 ) 1 M 378 (2, 2, 6 1 ) 2 M 378 (2, 6, 4) 2 M 388 (2, 4, 3) 1 M 388 (4, 4, 3) 1 M 388 (4, 4, 5 2 ) 1 M 388 (4, 6, 5 1 ) 1 M 388 (4, 2, 5 1 ) 1 M 388 (6, 4, 7 1 ) 1 M 388 (2, 2, 7 1 ) 1 M 388 (2, 2, 7 2 : 1) 1 M 388 (2, 2, 7 2 : 2) 2 M 388 (2, 4, 7 2 ) 1 M 388 (4, 4, 7 2 ) 1 M 388 (4, 4, 7 1 ) 1 M 388 (2, 4, 3) 1 M 388 (2, 4, 5 2 : 1) 1 M 388 (2, 4, 5 2 : 2) 2 M 388 (2, 2, 5 1 ) 1 M 388 (2, 6, 5 1 ) 1 M 388 (2, 6, 3 : 1) 2 M 388 (2, 6, 3 : 2) 2 2 M 388 (6, 6, 7 1 ) 2 M 388 (6, 4, 7 2 ) 2 M 388 (2, 6, 5 2 ) 2 M 555 (2, 2, 2) 1 M 555 (4 1, 2, 2) 1 M 555 (4 2, 2, 2) 1 M 555 (4 2, 4 1, 4 1 ) 1 M 555 (4 2, 4 1, 4 2 ) 2 M 556 (4 2, 1, 0 2 ) 1 M 556 (2, 1, 2) 1 M 556 (4 1, 0 2, 0 2 : 1) 1 M 556 (4 1, 0 2, 0 2 : 2) 2 M 556 (4 1, 1, 0 2 ) 1 M 556 (4 1, 2, 1) 1 M 556 (2, 0 2, 0 2 ) 1 M 556 (4 1, 2, 0 2 ) 2 M 556 (4 1, 2, 2) 2 M 556 (4 2, 1, 2) 2 M 556 (2, 0 1, 1) 2 M 556 (2, 0 2, 2) 2 M 556 (4 1, 0 1, 1) 3 M 556 (0, 2, 0 2 ) 2 3 M 556 (4 2, 2, 2) M 556 (0, 0 1, 0 1 ) M 557 (4 1, 4 2, 6) 1 M 557 (4 1, 4 2, 2) 1 M 557 (4 2, 2, 4 2 ) 1 M 557 (4 2, 6, 4 2 ) 1 M 558 (2, 2, 4 1 ) 1 M 558 (2, 6, 4 1 ) 1 M 558 (2, 4 2, 2) 1 M 558 (2, 4 2, 6) 1 M 558 (4 2, 4 1, 2) 1 M 558 (4 2, 6, 4 1 ) 1 M 558 (4 1, 6, 6 : 1) 1 M 558 (4 1, 6, 6 : 2) 2 M 558 (4 1, 4 2, 6) 1 M 558 (4 1, 2, 2 : 1) 1 M 558 (4 1, 2, 2 : 2) 2 M 558 (4 1, 4 2, 4 1 ) 1 M 558 (2, 6, 2 : 1) 1 M 558 (2, 6, 2 : 2) 2 M 558 (4 1, 6, 4 1 ) 1 M 558 (4 1, 2, 4 2 ) 1 M 558 (4 1, 2, 4 1 ) 1 M 558 (4 2, 4 2, 4 2 ) 1

23 On Rank 2 and Rank 3 Residually Connected Geometries for M M 558 (2, 6, 6) 2 M 558 (4 1, 6, 2) 2 M 558 (4 2, 6, 6) 2 M 558 (4 2, 2, 2) 2 M 558 (0, 6, 2) 2 2 M 558 (4 1, 4 2, 4 2 ) 2 M 558 (0, 4 2, 4 2 ) 3 M 558 (4 2, 2, 6) 2 2 M 566 (1, 0 2, 0 1 ) 1 M 566 (2, 1, 0 1 ) 1 M 566 (1, 1, 0 1 ) 1 M 566 (0 1, 1, 0 1 ) 2 M 566 (2, 0 2, 0 1 ) 2 M 566 (2, 2, 1) 2 2 M 566 (0 1, 0 2, 0 1 ) 2 2 M 566 (2, 2, 0 1 ) 2 2 M 566 (0 1, 0 1, 0 2 ) M 566 (0 1, 2, 0 2 ) M 567 (0 1, 6, 0) 2 2 M 567 (0 1, 4 1, 0) D 8 M 567 (0 2, 6, 0) 1 M 567 (0 2, 4 2, 0) 1 M 567 (0 2, 4 2, 2 2 ) 1 M 567 (0 2, 4 2, 2 1 ) 1 M 567 (0 2, 4 1, 0) 2 M 567 (0 2, 4 1, 2 1 ) 2 M 567 (0 2, 2, 2 1 ) 2 M 567 (0 2, 6, 2 1 ) 2 M 567 (1, 2, 0) 1 M 567 (1, 4 1, 0) 1 M 567 (1, 6, 2 1 ) 1 M 567 (1, 4 2, 0) 1 M 567 (1, 4 2, 2 2 ) 1 M 567 (1, 4 2, 2 1 ) 1 M 567 (1, 6, 0) 1 M 567 (1, 2, 2 1 ) 2 M 567 (1, 4 1, 2 1 ) 2 M 567 (2, 4 2, 1) 1 M 567 (2, 4 2, 2 1 ) 1 M 567 (2, 4 2, 0 : 1) 1 M 567 (2, 4 2, 0 : 2) 2 M 567 (2, 6, 2 1 ) 2 M 567 (2, 4 2, 2 2 ) 2 M 567 (2, 2, 0) 2 M 567 (2, 4 1, 0) 2 2 M 568 (0 1, 4, 0) 2 M 568 (0 1, 4, 2) 2 M 568 (0 1, 4, 1) 2 M 568 (0 1, 6, 1) 2 2 M 568 (0 1, 6, 0) 2 2 M 568 (0 1, 2, 2) 2 2 M 568 (0 2, 2, 1 2 : 1) 1 M 568 (0 2, 2, 1 2 : 2) 2 M 568 (0 2, 2, 2) 1 M 568 (0 2, 4 1, 1 1 ) 1 M 568 (0 2, 6, 0) 1 M 568 (0 2, 4 2, 0 : 1) 1 M 568 (0 2, 4 2, 0 : 2) 2 M 568 (0 2, 4 2, 1 1 ) 1 M 568 (0 2, 4 2, 1 2 ) 1 M 568 (0 2, 4 2, 2 : 1) 1 M 568 (0 2, 4 2, 2 : 2) 2 M 568 (0 2, 6, 2) 2 M 568 (0 2, 6, 1 1 ) 2 M 568 (0 2, 6, 1 2 ) 2 M 568 (0 2, 2, 1 1 ) 2 M 568 (0 2, 2, 0) 2 M 568 (1, 4 1, 0) 1 M 568 (1, 4 1, 2) 1 M 568 (1, 6, 2) 1 M 568 (1, 2, 2) 1 M 568 (1, 6, 0 : 1) 1 M 568 (1, 6, 0 : 2) 2

24 130 N. Kilic, P. Rowley M 568 (1, 4 2, 2) 1 M 568 (1, 2, 0) 2 M 568 (1, 2, 2) 2 M 568 (1, 4 2, 0) 1 M 568 (1, 6, 2) 2 M 568 (2, 4 2, 1 1 ) 1 M 568 (2, 4 1, 0) 1 M 568 (2, 4 1, 1 2 ) 1 M 568 (2, 4 1, 2) 1 M 568 (2, 4 2, 0) 2 M 568 (2, 4 2, 2) 2 M 568 (2, 6, 1 1 ) 2 M 568 (2, 2, 1 1 ) 2 M 568 (2, 2, 0) 2 M 568 (2, 6, 1 2 ) 2 M 568 (2, 6, 2) 2 M 577 (4 2, 4 2, 8 2 ) 1 M 577 (6, 4 2, 8 1 ) 1 M 577 (2, 6, 4) 1 M 577 (4 1, 4 2, 4) 1 M 577 (4 2, 6, 4) 1 M 577 (2, 2, 8 1 : 1) 1 M 577 (2, 2, 8 1 : 2) 2 M 577 (4 2, 4 2, 8 1 ) 1 M 577 (6, 6, 8 1 ) 1 M 577 (6, 4 1, 8 1 ) 1 M 577 (6, 2, 8 1 ) 2 M 577 (4 1, 6, 4) 2 M 577 (2, 4 2, 4) 2 M 577 (2, 4 1, 8 1 ) 2 M 577 (4 1, 4 1, 4) 2 2 M 578 (4 2, 4 1, 8) 1 M 578 (4 2, 4 2, 8) 1 M 578 (4 2, 6, 8) 1 M 578 (4 2, 2, 8) 1 M 578 (4 2, 4 1, 4) 1 M 578 (4 2, 2, 4) 1 M 578 (4 2, 6, 4) 1 M 578 (4 2, 4 2, 6 1 ) 1 M 578 (4 2, 4 1, 6 1 ) 1 M 578 (4 2, 4 2, 4) 1 M 588 (4 2, 4 1, 5 2 ) 1 M 588 (6, 6, 5 1 ) 1 M 588 (4 2, 6, 5 1 ) 1 M 588 (4 2, 4 1, 3) 1 M 588 (4 2, 4 2, 3) 1 M 588 (4 2, 4 1, 7 1 ) 1 M 588 (4 2, 4 2, 7 1 ) 1 M 588 (6, 4 1, 7 1 ) 1 M 588 (2, 2, 7 1 ) 1 M 588 (6, 6, 7 1 ) 1 M 588 (4 1, 4 2, 7 2 ) 1 M 588 (6, 6, 7 2 ) 1 M 588 (2, 2, 7 2 ) 1 M 588 (4 2, 4 2, 7 2 ) 1 M 588 (4 1, 2, 7 2 ) 1 M 588 (6, 4 1, 3) 1 M 588 (2, 6, 3 : 1) 1 M 588 (2, 6, 3 : 2) 2 M 588 (2, 4 1, 3) 1 M 588 (4 2, 2, 5 2 ) 1 M 588 (4 2, 6, 5 2 ) 1 M 588 (2, 2, 5 1 ) 1 M 588 (2, 6, 5 1 ) 1 M 588 (2, 2, 5 2 ) 2 M 588 (6, 6, 5 2 ) 2 M 588 (4 2, 2, 3) 2 M 588 (4 2, 6, 3) 2 M 588 (2, 6, 7 1 ) 2 M 588 (4 2, 6, 7 1 ) 2 M 588 (2, 4 2, 7 1 ) 2 M 666 (0 1, 0 1, 0 1 ) 1 M 666 (0 1, 0 1, 1) 1

25 On Rank 2 and Rank 3 Residually Connected Geometries for M M 666 (0 1, 1, 0 2 ) 2 M 666 (0 1, 0 2, 0 1 ) 2 M 666 (0 1, 0 2, 0 2 ) 2 2 M 666 (0 2, 0 2, 0 2 ) 2 3 M 667 (0 2, 0, 1) 1 M 667 (1, 0, 0) 1 M 667 (0 1, 0, 0 : 1) 1 M 667 (0 1, 0, 0 : 2) 2 M 667 (0 1, 0, 1) 1 M 667 (0 1, 0, 2 2 : 1) 1 M 667 (0 1, 0, 2 2 : 2) 2 M 667 (1, 2 2, 2 1 ) 2 M 667 (0 1, 2 1, 0) 2 M 667 (0 2, 2 1, 1) 2 M 667 (1, 2 1, 2 1 ) 2 M 667 (0 1, 2 2, 2 1 ) 2 M 667 (0 2, 0, 0) 2 2 M 668 (1, 0, 0 : 1) 1 M 668 (1, 0, 0 : 2) 2 M 668 (1, 2, 2 : 1) 1 M 668 (1, 2, 2 : 2) 2 M 668 (0 1, 1 1, 0) 1 M 668 (0 1, 2, 1 2 ) 1 M 668 (0 1, 1 2, 0) 1 M 668 (0 1, 1 1, 1 1 ) 1 M 668 (0 1, 0, 0) 1 M 668 (0 1, 0, 2) 1 M 668 (0 1, 1 2, 1 2 ) 1 M 668 (0 1, 1, 2) 1 M 668 (0 1, 2, 2) 1 M 668 (0 2, 1 1, 1 1 ) 2 M 668 (0 2, 0, 0) 2 M 668 (0 2, 1 2, 1 2 ) 2 M 668 (0 2, 1 1, 0) 2 M 668 (0 2, 1 1, 1 2 ) 2 M 668 (0 2, 0, 2 : 1) 2 M 668 (0 2, 0, 2 : 2) 2 2 M 668 (0 2, 2, 2) 2 M 668 (0 2, 1 1, 2) 2 M 668 (0 2, 1 2, 2) 2 2 M 677 (2 2, 1, 4) 1 M 677 (2 1, 0, 4) 1 M 677 (1, 0, 4) 1 M 677 (2 1, 0, 6) 1 M 677 (2 1, 2 1, 6) 1 M 677 (2 1, 2 2, 8 1 ) 1 M 677 (1, 2 2, 8 1 ) 1 M 677 (2 1, 1, 8 1 ) 1 M 677 (2 1, 0, 8 1 ) 1 M 677 (1, 0, 8 1 ) 1 M 677 (2 1, 2 1, 8 1 ) 1 M 677 (0, 0, 8 1 : 1) 1 M 677 (0, 0, 8 1 : 2) 2 M 677 (0, 0, 6) 1 M 677 (2 1, 2 1, 8 2 ) 2 M 677 (2 1, 1, 4) 2 M 677 (2 2, 2 2, 8 1 ) 2 M 677 (0, 0, 8 2 ) 2 M 677 (2 2, 0, 4 : 1) 2 M 677 (2 2, 0, 4 : 2) 2 2 M 677 (2 1, 2 1, 4) 2 M 678 (0, 1 1, 6 1 ) 1 M 678 (0, 1 2, 6 1 : 1) 1 M 678 (0, 1 2, 6 1 : 2) 2 M 678 (0, 0, 8 : 1) 1 M 678 (0, 0, 8 : 2) 2 M 678 (0, 1 1, 8) 1 M 678 (0, 1 1, 4) 1 M 678 (0, 2, 4 : 1) 1 M 678 (0, 2, 4 : 2) 2 M 678 (0, 0, 6 2 ) 1 M 678 (0, 1 1, 6 2 ) 1 M 678 (0, 2, 6 2 ) 1

26 132 N. Kilic, P. Rowley M 678 (0, 0, 6 1 ) 2 M 678 (0, 2, 6 1 ) 2 M 678 (0, 2, 8) 2 M 678 (0, 0, 4) 2 M 678 (2 1, 1 2, 6 1 ) 1 M 678 (2 1, 1 2, 8) 1 M 678 (2 1, 2, 8) 1 M 678 (2 1, 0, 6 2 ) 1 M 678 (2 1, 1 2, 6 2 ) 1 M 678 (2 1, 2, 6 2 ) 1 M 678 (2 1, 0, 4) 1 M 678 (2 1, 1, 4) 1 M 678 (2 1, 0, 6 1 ) 2 M 678 (2 1, 2, 6 1 ) 2 M 678 (2 1, 0, 8) 2 M 678 (2 1, 2, 4) 2 M 688 (0, 1 1, 5 2 ) 1 M 688 (0, 1 2, 5 2 ) 1 M 688 (1 2, 2, 5 2 ) 1 M 688 (1 1, 2, 5 2 ) 1 M 688 (0, 0, 7 2 ) 1 M 688 (1 2, 0, 7 2 ) 1 M 688 (1 1, 0, 7 2 ) 1 M 688 (2, 1 1, 7 2 ) 1 M 688 (2, 1 2, 7 2 ) 1 M 688 (2, 2, 7 2 ) 1 M 688 (0, 1 2, 3) 1 M 688 (1 2, 2, 3) 1 M 688 (0, 0, 7 1 ) 1 M 688 (1 2, 0, 7 1 ) 1 M 688 (1 1, 0, 7 1 ) 1 M 688 (2, 1 2, 7 1 ) 1 M 688 (2, 1 1, 7 1 ) 1 M 688 (2, 2, 7 1 ) 1 M 688 (0, 0, 5 1 ) 1 M 688 (2, 2, 5 1 ) 1 M 688 (0, 0, 5 2 ) 2 M 688 (2, 2, 5 2 ) 2 M 688 (1 1, 2, 3) 2 M 777 (8 1, 8 2, 8 1 ) 1 M 777 (8 1, 4, 8 1 ) 1 M 777 (4, 8 1, 4) 1 M 777 (4, 4, 8 2 ) 2 2 M 778 (8 1, 4, 6 1 ) 1 M 778 (8 1, 6 1, 6 1 : 1) 1 M 778 (8 1, 6 1, 6 1 : 2) 2 M 778 (8 1, 8, 6 1 ) 1 M 778 (4, 6 1, 6 1 ) 1 M 778 (8 1, 6 2, 6 1 ) 1 M 778 (8 1, 8, 8) 1 M 778 (4, 4, 8 : 1) 1 M 778 (4, 4, 8 : 2) 2 M 778 (8 1, 4, 4) 1 M 778 (8 1, 8, 4) 1 M 778 (4 1, 6 2, 8) 1 M 888 (3, 7 1, 5 1 ) 1 M 888 (3, 7 1, 3 : 1) 1 M 888 (3, 7 1, 3 : 2) 2 M 888 (3, 7 2, 3) 1 M 888 (3, 5 2, 5 1 ) 1 M 888 (5 1, 5 2, 5 2 ) 1 M 888 (5 2, 7 1, 7 1 ) 1 M 888 (5 2, 7 2, 5 2 ) 1 M 888 (7 1, 7 1, 7 2 ) 1 M 888 (7 1, 7 1, 3) 1 M 888 (5 2, 5 2, 3) 1 M 888 (7 1, 7 2, 7 2 ) 1 M 888 (5 2, 7 1, 5 2 ) 2 M 888 (3, 7 2, 7 2 ) 3 We now define the subgroups for M ijk (t i, t j, t k : l) (l {1, 2, 3}) by giving the X i X such that G i = Stab G X i (i = 1, 2, 3).

27 On Rank 2 and Rank 3 Residually Connected Geometries for M M ijk (t i, t j, t k : l) X 1 X 2 X 3 M 116 (0, 0, 0 : 1) {9} {22} {17, 11} M 116 (0, 0, 0 : 2) {19} {22} {17, 11} M 118 (0, 0, 1 : 1) {11} {22} M 118 (0, 0, 1 : 2) {1} {22} M 125 (0, 0, 2 : 1) {19} O 2 M 125 (0, 0, 2 : 2) {0} O 2 M 126 (0, 0, 0 : 1) {22} h 1 {17, 11} M 126 (0, 0, 0 : 2) {4} h 1 {17, 11} M 128 (0, 1, 3 : 1) {22} h 1 M 128 (0, 1, 3 : 2) {4} h 1 M 128 (0, 0, 3 : 1) {17} h 1 M 128 (0, 0, 3 : 2) {1} h 1 M 133 (0, 0, 3 : 1) {22} h 7 M 133 (0, 0, 3 : 2) {0} h 7 M 135 (0, 0, 2 : 1) {4} O 3 M 135 (0, 0, 2 : 2) {8} O 3 M 135 (1, 0, 2 : 1) {0} O 3 M 135 (1, 0, 2 : 2) {17} O 3 M 135 (0, 0, 4 : 1) {0} h 1 O 2 M 135 (0, 0, 4 : 2) {22} h 1 O 2 M 136 (0, 0, 1 : 1) {22} h 1 {17, 11} M 136 (0, 0, 1 : 2) {0} h 1 {17, 11} M 138 (0, 1, 2 : 1) {13} M 138 (0, 1, 2 : 2) {8} M 138 (1, 0, 2 : 1) {0} M 138 (1, 0, 2 : 2) {17} M 138 (0, 0, 4 : 1) {0} h 6 M 138 (0, 0, 4 : 2) {17} h 6 M 155 (0, 0, 4 1 : 1) {19} O 2 O 4 M 155 (0, 0, 4 1 : 2) {0} O 2 O 4 M 156 (0, 0, 2 : 1) {22} O 2 {17, 11} M 156 (0, 0, 2 : 2) {8} O 2 {17, 11} M 157 (0, 1, 4 2 : 1) {16}

28 134 N. Kilic, P. Rowley M 157 (0, 1, 4 2 : 2) {0} M 158 (0, 1, 2 : 1) {4} O 3 M 158 (0, 1, 2 : 2) {8} O 3 M 158 (0, 1, 4 1 : 1) {20} O 7 M 158 (0, 1, 4 1 : 2) {16} O 7 M 158 (0, 0, 4 1 : 1) {12} O 7 M 158 (0, 0, 4 1 : 2) {3} O 7 M 158 (0, 0, 6 : 1) {0} O 2 M 158 (0, 0, 6 : 2) {1} O 2 M 158 (0, 1, 6 : 1) {22} O 2 M 158 (0, 1, 6 : 2) {8} O 2 M 166 (0, 0, 0 1 : 1) {0} {17, 11} {22, 19} M 166 (0, 0, 0 1 : 2) {1} {17, 11} {22, 19} M 167 (0, 0, 1 : 1) {1} {22, 19} M 167 (0, 0, 1 : 2) {4} {22, 19} M 168 (0, 0, 0 : 1) {1} {17, 11} M 168 (0, 0, 0 : 2) {0} {17, 11} M 168 (0, 1, 0 : 1) {4} {17, 11} M 168 (0, 1, 0 : 2) {8} {17, 11} M 168 (0, 1, 0 : 3) {22} {17, 11} M 168 (0, 0, 2 : 1) {1} {22, 19} M 168 (0, 0, 2 : 2) {0} {22, 19} M 168 (0, 1, 2 : 1) {4} {22, 19} M 168 (0, 1, 2 : 2) {8} {22, 19} M 177 (1, 1, 8 1 : 1) {16} M 177 (1, 1, 8 1 : 2) {22} M 188 (0, 0, 7 1 : 1) {12} e 4 M 188 (0, 0, 7 1 : 2) {9} e 4 M 188 (1, 1, 7 1 : 1) {16} e 4 M 188 (1, 1, 7 1 : 2) {22} e 4

29 On Rank 2 and Rank 3 Residually Connected Geometries for M M 222 (2, 2, 2 : 1) h 5 M 222 (2, 2, 2 : 2) h 1 h 3 M 223 (2, 3, 1 : 1) h 5 h 5 M 223 (2, 3, 1 : 2) h 3 h 1 M 223 (2, 3, 3 : 1) h 1 h 3 h 1 M 223 (2, 3, 3 : 2) h 5 M 225 (2, 2, 2 : 1) h 3 O 4 M 225 (2, 2, 2 : 2) O 2 M 226 (2, 0, 0 : 1) h 1 {22, 19} M 226 (2, 0, 0 : 2) h 1 h 4 {17, 11} M 227 (2, 4, 4 : 1) h 3 h 4 M 227 (2, 4, 4 : 2) h 3 M 228 (2, 3, 3 : 1) h 1 h 3 e 1 M 228 (2, 3, 3 : 2) h 1 M 228 (2, 3, 3 : 3) h 1 h 3 M 228 (2, 3, 5 : 1) h 1 M 228 (2, 3, 5 : 2) h 1 M 228 (2, 1, 3 : 1)

30 136 N. Kilic, P. Rowley M 228 (2, 1, 3 : 2) h 1 h 6 M 233 (3, 3, 3 : 1) h 1 h 5 M 233 (3, 3, 3 : 1) h 1 h 5 M 233 (3, 3, 3 : 2) h 6 h 6 M 233 (1, 1, 3 : 1) h 1 h 5 h 7 M 233 (1, 1, 3 : 2) h 1 h 7 M 234 (3, 3, 2 : 1) h 1 h 1 M 234 (3, 3, 2 : 2) h 1 h 1 M 235 (3, 2, 2 : 1) h 1 h 1 M 235 (3, 2, 2 : 2) h 5 M 235 (3, 4, 4 : 1) h 1 h 6 M 235 (3, 4, 4 : 2) h 1 O 3 M 235 (1, 4, 2 : 1) h 6 O 7 M 235 (1, 4, 2 : 2) h 6 M 236 (3, 0, 0 : 1) h 1 h 6 {17, 13} M 236 (3, 0, 0 : 2) h 1 h 6 {17, 4} M 236 (3, 0, 0 : 3) h 1 h 6 {17, 11}

31 On Rank 2 and Rank 3 Residually Connected Geometries for M M 236 (1, 0, 0 : 1) h 6 {4, 19} M 236 (1, 0, 0 : 2) h 1 {0, 13} M 236 (1, 1, 0 : 1) h 6 {9, 19} M 236 (1, 1, 0 : 2) h 1 {22, 19} M 237 (3, 4, 4 : 1) h 3 h 1 M 237 (3, 4, 4 : 2) h 3 M 237 (3, 4, 2 : 1) h 3 h 9 M 237 (3, 4, 2 : 2) h 9 M 237 (1, 4, 2 : 1) h 9 M 237 (1, 4, 2 : 2) h 6 h 9 M 237 (1, 4, 4 : 1) h 6 M 237 (1, 4, 4 : 2) h 3 h 6 M 238 (3, 3, 4 : 1) h 5 M 238 (3, 3, 4 : 2) M 238 (3, 3, 2 : 1) h 5 M 238 (3, 3, 2 : 2) M 238 (3, 3, 2 : 3) h 4 M 238 (3, 1, 2 : 1) h 6

32 138 N. Kilic, P. Rowley M 238 (3, 1, 2 : 2) M 238 (1, 3, 2 : 1) M 238 (1, 3, 2 : 2) M 238 (1, 3, 4 : 1) h 6 M 238 (1, 3, 4 : 2) h 4 h 6 M 255 (2, 2, 4 1 : 1) O 2 M 255 (2, 2, 4 1 : 2) O 2 M 256 (2, 0, 1 : 1) O 2 {3, 4} M 256 (2, 0, 1 : 2) O 2 {8, 4} M 257 (2, 4, 6 : 1) M 257 (2, 4, 6 : 2) M 257 (2, 4, 4 1 : 1) O 3 M 257 (2, 4, 4 1 : 2) O 3 M 257 (2, 2, 4 2 : 1)

33 On Rank 2 and Rank 3 Residually Connected Geometries for M M 257 (2, 2, 4 2 : 2) M 258 (0, 3, 4 2 : 1) h 1 O 2 e 1 M 258 (0, 3, 4 2 : 2) h 3 O 3 e 1 M 258 (2, 3, 2 : 1) h 1 e 1 M 258 (2, 3, 2 : 2) O 3 M 258 (2, 3, 4 2 : 1) O 3 e 1 M 258 (2, 3, 4 2 : 2) O 3 e 1 M 258 (2, 3, 6 : 1) O 2 M 258 (2, 3, 6 : 2) O 2 M 258 (2, 1, 4 2 : 1) O 3 e 1 M 258 (2, 1, 4 2 : 2) h 5 O 2 e 1 M 258 (2, 5, 4 2 : 1) O 3 e 1 M 258 (2, 5, 4 2 : 2) O 2 e 1 M 258 (2, 5, 6 : 1) O 2

34 140 N. Kilic, P. Rowley M 258 (2, 5, 6 : 2) M 258 (4, 3, 4 1 : 1) M 258 (4, 3, 4 1 : 2) M 258 (4, 3, 6 : 1) O 7 O 7 O 2 M 258 (4, 3, 6 : 2) h 3 O 2 M 258 (4, 3, 2 : 1) O 3 M 258 (4, 3, 2 : 2) h 4 O 3 M 267 (0, 4, 0 : 1) h 3 {2, 19} M 267 (0, 4, 0 : 2) h 3 {7, 10} M 267 (0, 2, 0 : 1) h 5 {2, 16} M 267 (0, 2, 0 : 2) h 5 {4, 13} M 268 (0, 5, 0 : 1) {21, 12} M 268 (0, 5, 0 : 2) {11, 17} M 268 (1, 3, 0 : 1) h 1 {0, 12} M 268 (1, 3, 0 : 2) h 1 {0, 17} M 268 (1, 3, 2 : 1) h 1 {8, 16} M 268 (1, 3, 2 : 2) h 1 {4, 8} M 277 (2, 2, 8 1 : 1) h 5 M 277 (2, 2, 8 1 : 2)

35 On Rank 2 and Rank 3 Residually Connected Geometries for M M 277 (4, 2, 8 1 : 1) h 3 M 277 (4, 2, 8 1 : 2) M 278 (4, 1, 4 : 1) M 278 (4, 1, 4 : 2) h 6 M 278 (4, 3, 8 : 1) e 3 M 278 (4, 3, 8 : 2) h 4 e 3 M 278 (4, 5, 8 : 1) e 3 M 278 (4, 5, 8 : 2) h 6 e 3 M 288 (3, 5, 7 1 : 1) e 4 M 288 (3, 5, 7 1 : 2) e 4 M 288 (1, 3, 7 1 : 1) e 4 M 288 (1, 3, 7 1 : 2) e 4 M 333 (1, 1, 3 : 1) h 6 h 5 M 333 (1, 1, 3 : 2) h 6 h 5

36 142 N. Kilic, P. Rowley M 333 (3, 3, 3 : 1) h 1 h 7 M 333 (3, 3, 3 : 2) h 1 h 7 M 334 (3, 2, 2 : 1) h 1 h 5 M 334 (3, 2, 2 : 2) M 334 (1, 2, 2 : 1) h 5 h 8 M 334 (1, 2, 2 : 2) h 5 M 334 (1, 2, 4 : 1) h 5 M 334 (1, 2, 4 : 2) h 5 h 9 M 334 (3, 4, 4 : 1) h 5 M 334 (3, 4, 4 : 2) h 6 h 5 h 4 M 335 (1, 2, 4 : 1) h 5 O 3 M 335 (1, 2, 4 : 2) h 5 O 3 M 335 (3, 4, 2 : 1) h 1 O 2

37 On Rank 2 and Rank 3 Residually Connected Geometries for M M 335 (3, 4, 2 : 2) h 1 h 7 M 335 (3, 4, 4 : 1) h 1 O 2 M 335 (3, 4, 4 : 2) h 1 O 2 M 336 (3, 0, 1 : 1) M 336 (3, 0, 1 : 2) M 336 (1, 1, 0 : 1) M 336 (1, 1, 0 : 2) M 336 (1, 2, 0 : 1) M 336 (1, 2, 0 : 2) M 337 (3, 4, 2 : 1) h 1 h 1 h 1 h 6 h 1 h 7 {1, 17} h 7 {0, 17} h 5 {13, 17} {11, 13} h 5 {0, 22} {11, 17} h 9 M 337 (3, 4, 2 : 2) h 6 M 337 (3, 6, 4 : 1) h 5 M 337 (3, 6, 4 : 2) h 5 M 337 (3, 2, 2 : 1) h 9 M 337 (3, 2, 2 : 2) h 9 M 337 (1, 4, 2 : 1) h 6

38 144 N. Kilic, P. Rowley M 337 (1, 4, 2 : 2) h 6 M 338 (3, 2, 4 : 1) e 1 M 338 (3, 2, 4 : 2) M 338 (3, 2, 2 : 1) M 338 (3, 2, 2 : 2) e 1 M 338 (1, 2, 4 : 1) M 338 (1, 2, 4 : 2) M 345 (2, 2, 4 : 1) O 3 M 345 (2, 2, 4 : 2) h 8 O 3 M 345 (2, 4, 4 : 1) h 4 O 7 M 345 (2, 4, 4 : 2) M 346 (2, 0, 0 : 1) M 346 (2, 0, 0 : 2) M 346 (2, 1, 0 : 1) h 8 O 7 h 10 {8, 20} h 10 {8, 5} h 10 {5, 19}

39 On Rank 2 and Rank 3 Residually Connected Geometries for M M 346 (2, 1, 0 : 2) h 10 {9, 19} M 347 (4, 2, 2 : 1) M 347 (4, 2, 2 : 2) h 10 M 347 (2, 6, 4 : 1) M 347 (2, 6, 4 : 2) M 347 (2, 2, 4 : 1) M 347 (2, 2, 4 : 2) h 9 M 348 (4, 4, 5 : 1) h 4 M 348 (4, 4, 5 : 2) h 4 M 348 (4, 2, 3 : 1) h 3 M 348 (4, 2, 3 : 2) h 3

40 146 N. Kilic, P. Rowley M 348 (4, 4, 3 : 1) h 3 M 348 (4, 4, 3 : 2) h 3 M 348 (4, 6, 5 : 1) h 4 M 348 (4, 6, 5 : 2) h 1 h 4 M 348 (4, 2, 1 : 1) M 348 (4, 2, 1 : 2) M 348 (0, 2, 5 : 1) h 4 M 348 (0, 2, 5 : 2) h 4 M 348 (2, 4, 5 : 1) h 4 M 348 (2, 4, 5 : 2) h 6 h 10 M 348 (2, 4, 3 : 1) h 6 M 348 (2, 4, 3 : 2) h 3 M 348 (2, 6, 3 : 1) h 1

41 On Rank 2 and Rank 3 Residually Connected Geometries for M M 348 (2, 6, 3 : 2) h 3 M 348 (2, 2, 3 : 1) e 1 M 348 (2, 2, 3 : 2) M 348 (2, 4, 1 : 1) h 5 h 8 M 348 (2, 4, 1 : 2) h 6 M 355 (2, 2, 4 1 : 1) O 2 O 4 M 355 (2, 2, 4 1 : 2) O 2 O 4 M 355 (2, 2, 2 : 1) O 2 M 355 (2, 2, 2 : 2) O 2 M 355 (2, 2, 4 1 : 1) O 2 O 4 M 355 (2, 2, 4 1 : 2) O 2 O 4

42 148 N. Kilic, P. Rowley M 356 (2, 0, 0 2 : 1) O 2 {9, 20} M 356 (2, 0, 0 2 : 2) O 2 {5, 6} M 356 (2, 1, 0 1 : 1) O 2 {1, 22} M 356 (2, 1, 0 1 : 2) O 2 {0, 8} M 357 (2, 2, 6 : 1) M 357 (2, 2, 6 : 2) M 358 (4, 2, 4 2 : 1) O 5 M 358 (4, 2, 4 2 : 2) O 4 M 358 (4, 2, 2 : 1) O 3 M 358 (4, 2, 2 : 2) M 358 (4, 4, 2 : 1) O 3 M 358 (4, 4, 2 : 2) h 6 M 366 (1, 0, 0 1 : 1) {11, 17} {22, 19} M 366 (1, 0, 0 1 : 2) {11, 17} {19, 22} M 366 (0, 0, 0 1 : 1) {11, 17} {22, 19}

43 On Rank 2 and Rank 3 Residually Connected Geometries for M M 366 (0, 0, 0 1 : 2) {11, 17} {22, 19} M 367 (1, 2, 0 : 1) {1, 13} M 367 (1, 2, 0 : 2) {1, 19} M 367 (1, 4, 2 1 : 1) h 6 {8, 17} M 367 (1, 4, 2 1 : 2) h 6 {8, 9} M 367 (0, 4, 0 : 1) h 1 {1, 19} M 367 (0, 4, 0 : 2) h 6 {4, 13} M 367 (0, 2, 1 : 1) {22, 19} M 367 (0, 2, 1 : 2) {22, 19} M 368 (0, 4, 2 : 1) h 6 {4, 19} M 368 (0, 4, 2 : 2) h 6 {4, 13} M 368 (0, 4, 0 : 1) h 6 {0, 17} M 368 (0, 4, 0 : 2) h 6 {5, 6} M 368 (0, 2, 2 : 1) {18, 13} M 368 (0, 2, 2 : 2) {4, 8} M 368 (0, 2, 2 : 3) {4, 13} M 368 (0, 2, 0 : 1) {5, 6} M 368 (0, 2, 0 : 2) {1, 9} M 368 (1, 2, 0 : 1) {1, 11} M 368 (1, 2, 0 : 2) {1, 0} M 368 (1, 4, 0 : 1) h 6 {1, 17} M 368 (1, 4, 0 : 2) h 6 {1, 9} M 368 (2, 2, 0 : 1) {17, 0} M 368 (2, 2, 0 : 2) {3, 15} M 377 (2, 4, 4 : 1) d 3

44 150 N. Kilic, P. Rowley M 377 (2, 4, 4 : 2) d 3 M 378 (6, 4, 8 : 1) M 378 (6, 4, 8 : 2) e 1 e 3 M 378 (6, 2, 4 : 1) M 378 (6, 2, 4 : 2) M 378 (2, 2, 8 : 1) e 1 M 378 (2, 2, 8 : 2) e 3 M 378 (2, 2, 4 : 1) e 1 M 378 (2, 2, 4 : 2) e 1 M 378 (2, 4, 8 : 1) e 3 M 378 (2, 4, 8 : 2) e 1

45 On Rank 2 and Rank 3 Residually Connected Geometries for M M 388 (2, 2, 7 2 : 1) e 1 M 388 (2, 2, 7 2 : 2) e 1 M 388 (2, 4, 5 2 : 1) M 388 (2, 4, 5 2 : 2) e 1 M 388 (2, 6, 3 : 1) h 9 e 3 M 388 (2, 6, 3 : 2) e 3 M 556 (4 1, 0 2, 0 2 : 1) O 2 O 4 {0, 19} M 556 (4 1, 0 2, 0 2 : 2) O 2 O 4 {9, 19} M 558 (4 1, 6, 6 : 1) O 6 e 1 M 558 (4 1, 6, 6 : 2) O 2 M 558 (4 1, 2, 2 : 1) O 5 e 1 M 558 (4 1, 2, 2 : 2) O 3 M 558 (2, 6, 2 : 1) O 2

46 152 N. Kilic, P. Rowley M 558 (2, 6, 2 : 2) O 6 e 1 M 567 (2, 4 2, 0 : 1) {12, 16} M 567 (2, 4 2, 0 : 2) {12, 21} M 568 (0 2, 2, 1 2 : 1) O 4 {3, 4} M 568 (0 2, 2, 1 2 : 2) O 5 {13, 19} e 1 M 568 (0 2, 4 2, 0 : 1) O 3 {3, 0} e 1 M 568 (0 2, 4 2, 0 : 2) O 2 {1, 19} e 1 M 568 (0 2, 4 2, 2 : 1) O 3 {8, 20} e 1 M 568 (0 2, 4 2, 2 : 2) O 2 {9, 22} e 1 M 568 (1, 6, 0 : 1) O 2 {1, 17} M 568 (1, 6, 0 : 2) O 2 {0, 17} M 577 (2, 2, 8 1 : 1) M 577 (2, 2, 8 1 : 2) M 588 (2, 6, 3 : 1) e 3 M 588 (2, 6, 3 : 2) O 6 e 1 M 667 (0 1, 0, 0 : 1) {4, 13} {19, 16} M 667 (0 1, 0, 0 : 2) {4, 13} {2, 7} M 667 (0 1, 0, 2 2 : 1) {4, 13} {17, 22} M 667 (0 1, 0, 2 2 : 2) {17, 11} {10, 16}

47 On Rank 2 and Rank 3 Residually Connected Geometries for M M 668 (1, 0, 0 : 1) {11, 19} {1, 19} e 1 M 668 (1, 0, 0 : 2) {11, 17} {0, 17} M 668 (1, 2, 2 : 1) {17, 22} {9, 22} e 1 M 668 (1, 2, 2 : 2) {19, 22} {8, 22} M 668 (0 2, 0, 2 : 1) {11, 17} {8, 4} M 668 (0 2, 0, 2 : 2) {11, 17} {13, 4} M 677 (0, 0, 8 1 : 1) {19, 21} M 677 (0, 0, 8 1 : 2) {19, 1} M 677 (2 2, 0, 4 : 1) {5, 22} d 3 M 677 (2 2, 0, 4 : 2) {9, 22} d 3 M 678 (0, 1 2, 6 1 : 1) {0, 7} M 678 (0, 1 2, 6 1 : 2) {0, 8} M 678 (0, 0, 8 : 1) {13, 19} e 3 M 678 (0, 0, 8 : 2) {13, 4} e 3 M 678 (0, 2, 4 : 1) {13, 4} e 3 M 678 (0, 2, 4 : 2) {13, 19} M 778 (8 1, 6 1, 6 1 : 1) M 778 (8 1, 6 1, 6 1 : 2) M 778 (4, 4, 8 : 1) M 778 (4, 4, 8 : 2)