Báo cáo toán học: "Rank three residually connected geometries for M22, revisited" pdf

Rank three residually connected geometries for M 22 , revisited Dimitri Leemans Universite Libre de Bruxelles Departement de Mathematiques Service de Geometrie - CP 216 Boulevard du Triomphe B-1050 Bruxelles Belgium dleemans@ulb.ac.be Peter Rowley School of Mathematics University of Manchester Oxford Road Manchester M13 6PL, UK peter.j.rowley@manchester.ac.uk Submitted: Aug 20, 2009; Accepted: Dec 22, 2009; Published: Jan 5, 2010 Mathematics Subject Classifications: 20D08, 51E10, 05C25 Abstract The rank 3 residually connected flag transitive geometries Γ for M 22 for which the stabilizer of each object in Γ is a maximal subgroup of M 22 are determined. As a result this deals with the infelicities in Theorem 3 of Kilic and Rowley, On rank 2 and rank 3 residually connected geometries for M 22 . Note di Matematica, 22(2003), 107–154. 1 Introduction Here we report on calculations carried out using Magma[2] on certain rank 3 geometries for M 22 , the Mathieu group of degree 22. Putting G = M 22 , the main conclusion is as follows. Theorem 1. Up to conjugacy in Aut(G) there ar e 431 rank 3 residually connected flag transitive geometries Γ satisfying the condition that Stab G (x) is a maximal subgroup of G for all x ∈ Γ. These 431 geometries are tabulated in Section 2 where they are described in terms of the action of M 22 on a 22-element set. These geometries may also be downloaded f r om [6]. This list supersedes that given in Theorem 3 of [5], which not only omits some of the geometries but also contains geometries that should not be there (usually because they fail to be flag transitive). We next introduce some not ation, mostly for use in describing the geometries in Section 2. Our notation for geometries is standard, as may be found in the electronic journal of combinatorics 17 (2010), #N4 1 [1]. So a geometry Γ consists of a triple (Γ, I, ⋆) where Γ is a set, I the set of types and ⋆ a symmetric incidence relation on Γ for which (i) Γ = . ∪ i∈I Γ i with each Γ i a non-empty subset of Γ; and (ii) if x ∈ Γ i , y ∈ Γ j (i, j ∈ I) and x ⋆ y, then i = j. The rank of Γ, rank Γ, is the cardinality of I – if |I| = n we shall take I = {1, , n}. If F ⊆ Γ has the property that for all x, y ∈ Γ with x = y, we have x ⋆ y, then we call F a flag of Γ. The rank of F is just |F |, its corank is |{i ∈ I|F ∩ Γ i = ∅}| and its type is {i ∈ I|F ∩ Γ i = ∅}.The geometries we consider will be assumed to possess at least one flag of rank |I|. Let H be a subgroup of the group of automorphisms of Γ, AutΓ, which consists of all permutations of Γ preserving the sets Γ i and the incidence relation. By saying that Γ is a flag transitive geometry for H we mean that if F 1 and F 2 are flags of Γ which have the same type, then F 1 h = F 2 for some h ∈ H. Assume that Γ is a flag transitive geometry for H, and let F = {x 1 , , x n } be a maximal flag of Γ (t hat is F has rank |I|). For i ∈ I set H i = Stab H (x i ) and for ∅ = J ⊆ I set H J = ∩ j∈J H x j and if J = {i 1 , , i j } we also write H J as H i 1 i j .A geometry Γ is said to be residually connected if for all flags F of Γ of corank at least 2, the incidence graph of Γ F = {x ∈ Γ|y ⋆ x for all y ∈ F } is connected. For the remainder of this paper G will denote M 22 , the Mathieu group of degree 2 2 and Ω a 24-element set whose elements will be labelled as in Curtis [4 ]. So Ω = 15 3 0 ∞ 18 20 8 14 10 16 4 17 2 7 13 11 21 12 1 22 6 5 9 19 and we use the MOG [4] to give us a Steiner system S(24, 8, 5) for Ω. Further we shall identify G with Stab M 24 (∞) ∩ Stab M 24 (14) where M 24 is the Mathieu group of degree 24 leaving invariant the Steiner system on Ω given by the MOG . Set Λ = Ω \ {∞, 14}. An 8-element block of the Steiner system is referred to as an octad of Ω and a dodecad is the symmetric difference of two octads of Ω which intersect in a set of size two. We shall follow the notation in [5]. So H={X ⊆ Λ|X ∪ {∞ , 14} is an octad of Ω} (hexads of Λ), H p ={X ⊆ Λ|X ∪ {14} is an octad of Ω} (heptads o f Λ), H p ∞ ={X ⊆ Λ|X ∪ {∞} is an octad of Ω} (heptads of Λ), O={X ⊆ Λ|X is an octad of Ω} (octads of Λ), D={X ⊆ Λ||X| = 2} (duads of Λ). D o ={X ⊆ Λ|X is a dodecad of Ω} (dodecads of Λ) and E={X ⊆ Λ| one of X ∪ {∞} and X ∪ {14} is a dodecad o f Ω} (endecads of Λ). Set X= Λ∪H∪H p ∪H p ∞ ∪O∪D∪D o ∪E. Up to conjugacy, the maximal subgroups M i of G are as follows (see [3]). the electronic journal of combinatorics 17 (2010), #N4 2 M i Description M 1 ∼ = L 3 (4) M 1 = Stab G (a), a ∈ Λ M 2 ∼ = 2 4 : A 6 M 2 = Stab G (X), X ∈ H M 3 ∼ = A 7 M 3 = Stab G (X), X ∈ H p M 4 ∼ = A 7 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 ∼ = M 10 M 7 = Stab G (X), X ∈ D o M 8 ∼ = L 2 (11) M 8 = Stab G (X), X ∈ E For i ∈ {1, , 8}, M i will denote the G-conjugacy class of M i . We observe that if X, Y ∈ X \ E, then we have X = Y if and only if Stab G (X) = Stab G (Y ). The description of the geometries listed in Section 2 follows the following scheme. By Γ(G, {G i , G j , G k }) =M abc (t ab , t ac , t bc ) we mean that G i = Stab G (X i ) ∈ M a , G j = Stab G (X j ) ∈ M b , G k = Stab G (X k ) ∈ M c with |X i ∩ X j | = t ab , |X i ∩ X k | = t ac and |X j ∩ X k | = t bc . Some care is needed with this notation when, say X k ∈ E (as t hen we have two endecads to choose from, namely X k and Λ\X k ). So, concerning Theorem 2 o f [5], M 18 (1) and M 18 (0) describe the same geometry (up to Aut(G) conjugacy). Just as M 28 (5) and M 28 (1) are the same (up to Aut(G) conjugacy). With the removal of duplicates such as those mentioned, that is M 18 (1), M 28 (5), M 58 (6), M 68 (2) and M 78 (8), the list in Theorem 2 of [5] is correct (so yielding that there are 81 such rank 2 geometries). For the meaning of subscripts (such as, for example, M 155 (1, 0, 4 1 )) and the number fo llowing a colon (as in M 378 (2, 2, 4 : 2) ) we refer the r eader to [5]. Finally the column in the tables called Number gives the number of the geometry in [6]. There the geometries are given as an ordered sequence called geo – so f or 1  j  431, {G 1 , G 2 , G 3 } = {geo[j][1], geo[j][2], geo[j][3]}. 2 The Rank Three Geometries Γ Numb er G 123 Γ Numb er G 123 M 111 (0, 0, 0) 31 2 4 3 M 112 (0, 0, 0) 34 S 4 M 112 (0, 0, 1) 14 A 5 M 112 (0, 1, 1) 12 2 4 A 4 M 113 (0, 0, 0) 36 A 4 M 113 (0, 0, 1) 37 S 4 M 113 (0, 1, 1) 32 A 5 M 115 (0, 0, 1) 38 A 4 M 115 (0, 1, 1) 33 S 4 M 116 (0, 1, 0) 2 2 4 3 M 117 (0, 1, 0) 40 S 3 M 117 (0, 0, 0) 35 Q 8 M 118 (0, 1, 1) 39 S 3 *********** **** **** *********** **** **** M 122 (0, 0, 0) 42 3 2 4 M 122 (0, 1, 0) 13 A 5 M 122 (0, 0, 2) 49 D 8 M 122 (1, 0, 2) 18 S 4 M 122 (1, 1, 2) 15 2 2 A 4 M 123 (0, 0, 1) 57 S 3 M 123 (0, 1, 1) 51 D 10 M 123 (1, 0, 1) 27 A 4 M 123 (0, 0, 3) 56 S 3 M 123 (0, 1, 3) 47 3 2 2 M 123 (1, 0, 3) 19 S 4 M 123 (1, 1, 3) 20 S 4 the electronic journal of combinatorics 17 (2010), #N4 3 Γ Number G 123 Γ Numb er G 123 M 125 (0, 0, 0) 44 S 4 M 125 (0, 1, 0) 45 S 4 M 125 (1, 0, 0) 16 2 2 D 8 M 125 (0, 1, 2) 53 4 M 125 (1, 0, 2) 28 S 3 M 125 (1, 1, 2) 21 A 4 M 125 (0, 0, 4) 48 D 8 M 125 (0, 1, 4) 43 S 4 M 125 (1, 1, 4) 17 S 4 M 126 (1, 0, 0) 24 2 3 M 126 (0, 1, 0) 5 S 4 M 126 (0, 0, 1) 54 2 2 M 126 (1, 0, 1) 22 A 4 M 126 (0, 0, 2) 46 S 4 M 126 (1, 1, 2) 1 2 4 A 4 M 127 (0, 0, 2) 52 4 M 127 (1, 0, 2) 26 S 3 M 127 (0, 0, 4) 59 2 M 127 (0, 1, 4) 58 2 M 127 (1, 1, 4) 29 2 2 M 127 (1, 0, 4) 23 Q 8 M 127 (0, 0, 6) 41 3 2 4 M 128 (0, 1, 1) 55 S 3 M 128 (0, 0, 1) 50 D 10 M 128 (1, 0, 1) 25 A 4 M 128 (1, 0, 3) 30 2 2 *********** **** **** *********** **** **** M 133 (0, 0, 1) 115 4 M 133 (1, 0, 1) 73 S 3 M 133 (0, 3, 1) 72 S 3 M 133 (1, 1, 3) 68 D 8 M 134 (0, 0, 0) 108 F 21 M 134 (1, 0, 0) 61 S 4 M 134 (0, 0, 2) 118 2 M 134 (0, 1, 2) 77 4 M 134 (1, 1, 2) 67 D 10 M 134 (0, 0, 4) 112 S 3 M 134 (1, 1, 4) 64 3 2 2 M 134 (1, 0, 4) 60 S 4 M 135 (0, 1, 0) 83 F 21 M 135 (0, 0, 0) 102 S 4 M 135 (1, 0, 0) 61 S 4 M 135 (0, 1, 2) 100 2 M 135 (1, 1, 2) 74 S 3 M 135 (1, 1, 4) 75 S 3 M 135 (0, 1, 4) 92 S 3 M 135 (1, 0, 4) 69 D 8 M 136 (0, 1, 0) 9 A 4 M 136 (1, 0, 1) 78 2 2 M 136 (0, 0, 2) 105 D 8 M 136 (1, 0, 2) 63 S 4 M 136 (1, 1, 2) 3 A 5 M 137 (0, 1, 2) 117 2 M 137 (0, 0, 2) 114 4 M 137 (1, 0, 2) 76 4 M 137 (1, 1, 2) 66 D 10 M 137 (1, 0, 4) 80 2 M 137 (0, 0, 6) 113 4 M 137 (0, 1, 6) 111 S 3 M 137 (1, 1, 6) 71 S 3 M 138 (0, 0, 2) 116 2 M 138 (1, 1, 2) 70 S 3 M 138 (1, 0, 4) 79 2 M 138 (0, 0, 6) 110 S 3 M 138 (1, 1, 6) 65 D 10 M 138 (0, 1, 6) 109 A 4 *********** **** **** *********** **** **** M 155 (0, 1, 0) 81 S 4 M 155 (1, 1, 2) 90 4 M 155 (1, 0, 4 1 ) 93 3 M 155 (1, 1, 4 1 ) 94 3 M 155 (0, 1, 4 2 ) 85 D 8 M 155 (1, 1, 4 2 ) 84 D 8 M 156 (0, 0, 0 1 ) 101 2 2 2 2 M 156 (1, 0, 0 1 ) 82 S 4 M 156 (1, 0, 0 2 ) 95 2 M 156 (0, 1, 0 2 ) 8 2 3 M 156 (1, 0, 2) 86 2 3 M 156 (1, 1, 2) 4 S 4 the electronic journal of combinatorics 17 (2010), #N4 4 Γ Number G 123 Γ Numb er G 123 M 157 (1, 0, 2) 89 4 M 157 (0, 0, 2) 103 S 3 M 157 (1, 0, 4 1 ) 104 4 M 157 (1, 0, 4 1 ) 87 S 3 M 157 (1, 0, 4 2 ) 99 1 M 157 (1, 1, 4 2 ) 98 1 M 157 (1, 0, 6) 88 4 M 158 (1, 0, 2) 97 2 M 158 (1, 0, 4 1 ) 96 2 M 158 (1, 1, 2) 91 S 3 *********** **** **** *********** **** **** M 166 (1, 0, 0 1 ) 10 S 3 M 167 (0, 0, 0) 107 2 M 167 (1, 0, 0) 7 Q 8 M 167 (0, 1, 2 1 ) 106 2 M 167 (1, 1, 2 1 ) 6 D 10 M 168 (1, 0, 0) 11 S 3 *********** **** **** *********** **** **** M 177 (0, 1, 4) 121 2 M 177 (1, 1, 4) 125 2 2 M 177 (0, 0, 4) 119 Q 8 M 177 (1, 0, 8 1 ) 124 1 M 177 (0, 0, 8 2 ) 120 4 M 178 (0, 0, 6 1 ) 123 2 M 178 (0, 0, 4) 122 2 M 188 (0, 0, 3) 126 2 2 M 188 (0, 0, 5 2 ) 128 2 M 188 (1, 0, 7 2 ) 127 3 *********** **** **** *********** **** **** M 222 (0, 2, 2) 162 D 8 M 222 (0, 0, 2) 159 S 4 M 223 (0, 3, 1) 170 S 3 M 223 (0, 1, 1) 164 D 10 M 223 (0, 3, 3) 160 3 2 2 M 225 (0, 2, 2) 168 2 M 225 (0, 4, 2) 165 D 8 M 225 (0, 0, 4) 158 S 4 M 225 (2, 2, 0) 181 2 2 M 225 (2, 0, 0) 173 2 4 2 M 226 (0, 0, 1) 166 S 3 M 226 (0, 0, 0) 161 D 8 M 226 (0, 0, 2) 130 S 4 M 226 (2, 1, 0) 202 2 M 226 (2, 1, 2) 135 A 4 M 226 (2, 0, 2) 133 2 × D 8 M 227 (2, 6, 4) 154 D 8 M 227 (0, 4, 2) 171 2 M 227 (0, 4, 4) 167 2 M 228 (0, 3, 5) 169 S 3 M 228 (0, 1, 5) 163 D 10 M 228 (2, 1, 1) 201 S 3 M 228 (2, 5, 1) 200 A 4 *********** **** **** *********** **** **** M 233 (1, 1, 1) 241 2 M 234 (1, 1, 0) 236 A 4 M 234 (3, 1, 0) 226 S 3 M 234 (3, 1, 4) 225 S 3 M 235 (3, 2, 0) 227 S 3 M 235 (3, 0, 0) 175 S 4 M 235 (1, 0, 4) 185 2 2 M 235 (1, 0, 2) 186 2 2 M 235 (1, 2, 0) 237 S 3 M 235 (1, 4, 4) 195 S 3 M 235 (1, 4, 0) 190 A 4 M 236 (1, 2, 0) 145 S 3 M 236 (3, 1, 0) 213 2 M 236 (3, 1, 2) 206 S 3 M 236 (3, 2, 1) 141 D 8 M 236 (3, 0, 2) 216 D 12 M 236 (3, 2, 2) 134 S 4 M 236 (1, 0, 2) 218 2 2 M 237 (3, 2, 6) 224 S 3 M 237 (3, 6, 6) 151 3 2 2 M 237 (1, 4, 6) 240 2 M 237 (1, 2, 2) 239 2 the electronic journal of combinatorics 17 (2010), #N4 5 Γ Number G 123 Γ Number G 123 M 237 (1, 6, 2) 155 D 10 M 238 (3, 5, 6) 222 S 3 M 238 (1, 3, 6) 238 2 M 238 (1, 1, 2) 228 2 2 M 238 (3, 1, 5) 223 S 3 *********** **** **** *********** **** **** M 255 (0, 2, 4 1 ) 187 2 M 255 (4, 4, 4 1 ) 196 3 M 255 (2, 2, 0) 243 2 2 M 255 (2, 4, 4 2 ) 193 2 2 M 255 (4, 0, 2) 179 2 2 M 255 (0, 0, 4 2 ) 174 2 4 M 255 (4, 0, 0) 172 2 4 2 M 256 (0, 0, 1) 182 3 M 256 (0, 1, 0 2 ) 180 2 2 M 256 (0, 0, 0 2 ) 176 2 3 M 256 (0, 2, 0 1 ) 129 2 4 2 2 M 256 (2, 1, 2) 208 2 M 256 (2, 2, 1) 146 3 M 256 (2, 2, 0 2 ) 142 2 2 M 256 (2, 1, 0 1 ) 203 S 3 M 256 (2, 0, 0 1 ) 215 D 8 M 256 (4, 0, 1) 197 2 M 256 (4, 1, 0 1 ) 194 2 2 M 256 (4, 1, 2) 191 S 3 M 256 (4, 2, 2) 131 2 2 2 2 M 256 (4, 0, 0 1 ) 188 2 2 2 2 M 257 (0, 2, 4 2 ) 184 2 M 257 (0, 4, 6) 177 2 2 M 257 (0, 4, 2) 178 2 2 M 257 (2, 6, 4 2 ) 157 2 M 257 (4, 4, 2) 189 D 8 M 257 (4, 4, 4 1 ) 192 2 2 M 258 (0, 5, 2) 183 2 2 M 258 (4, 3, 4 2 ) 199 1 M 258 (4, 1, 4 1 ) 198 2 M 258 (2, 5, 6) 234 2 *********** **** **** *********** **** **** M 266 (1, 1, 0 1 ) 214 1 M 266 (1, 1, 0 2 ) 204 2 2 M 266 (2, 0, 0 1 ) 143 2 2 M 266 (1, 0, 0 2 ) 205 2 2 M 266 (2, 0, 0 2 ) 132 2 × D 8 M 266 (2, 1, 0 1 ) 137 S 3 M 267 (0, 4, 2 1 ) 221 1 M 267 (0, 6, 0) 153 D 8 M 267 (1, 2, 0) 211 1 M 267 (1, 4, 2 2 ) 212 1 M 267 (1, 4, 0) 210 1 M 267 (1, 2, 2 1 ) 207 2 M 267 (1, 6, 2 1 ) 150 D 10 M 267 (2, 4, 1) 149 2 M 267 (2, 2, 0) 139 2 2 M 267 (2, 4, 2 1 ) 140 4 M 267 (2, 4, 2 2 ) 136 D 8 M 268 (0, 1, 1 1 ) 219 2 M 268 (0, 1, 1 2 ) 220 2 M 268 (0, 1, 0) 217 2 2 M 268 (1, 1, 0) 209 2 M 268 (2, 3, 1 1 ) 147 2 M 268 (2, 3, 0) 148 2 2 M 268 (2, 3, 1 2 ) 138 2 2 M 268 (2, 1, 0) 144 S 3 *********** **** **** *********** **** **** M 277 (4, 6, 8 1 ) 156 2 M 277 (2, 2, 4) 242 2 2 M 277 (2, 6, 4) 152 D 8 M 278 (4, 5, 4) 233 2 M 278 (4, 5, 3) 232 2 *********** **** **** *********** **** **** M 288 (5, 1, 5 1 ) 235 1 M 288 (1, 3, 7 2 ) 230 2 M 288 (3, 5, 7 2 ) 231 2 M 288 (1, 1, 7 2 ) 229 3 *********** **** **** *********** **** **** the electronic journal of combinatorics 17 (2010), #N4 6 Γ Number G 123 Γ Numb er G 123 M 333 (1, 1, 1) 422 2 M 334 (3, 2, 0) 405 2 M 334 (1, 2, 0) 402 4 M 334 (1, 4, 0) 400 S 3 M 334 (3, 0, 0) 399 A 4 M 334 (1, 4, 4) 406 3 2 2 M 335 (3, 0, 2) 371 2 M 335 (1, 4, 4) 394 2 M 335 (1, 2, 0) 361 S 3 M 335 (1, 0, 4) 364 4 M 336 (3, 0, 1 : 2) 346 2 M 336 (3, 2, 1) 294 2 M 336 (3, 2, 0) 285 2 2 M 336 (3, 2, 2) 278 D 8 M 337 (3, 6, 6) 420 2 M 337 (3, 6, 2) 416 4 M 337 (1, 2, 2) 421 4 M 338 (1, 6, 4) 412 2 M 338 (3, 6, 2) 409 2 2 M 338 (3, 6, 6) 408 S 3 *********** **** **** *********** **** **** M 345 (2, 2, 0) 372 2 M 345 (0, 4, 2) 393 2 M 345 (2, 4, 0) 365 4 M 345 (4, 0, 2) 362 S 3 M 345 (0, 0, 4) 358 A 4 M 345 (4, 0, 0) 357 S 4 M 346 (4, 1, 2) 282 S 3 M 346 (4, 2, 2) 276 D 12 M 346 (0, 0, 0) 347 S 3 M 346 (0, 2, 0) 279 D 8 M 346 (0, 0, 1) 344 3 M 346 (2, 2, 1) 295 2 M 346 (2, 2, 0) 296 2 M 347 (0, 2, 4) 404 2 M 347 (0, 5, 1) 401 4 M 347 (2, 6, 6) 417 4 M 348 (2, 1, 6) 407 D 10 M 348 (0, 2, 3) 403 2 M 348 (0, 6, 1) 398 A 4 M 348 (2, 2, 1) 413 2 *********** **** **** *********** **** **** M 355 (0, 2, 2) 374 2 M 355 (0, 2, 4 1 ) 375 2 M 355 (0, 4, 2) 373 2 M 355 (0, 2, 4 2 ) 366 2 2 M 355 (4, 2, 0) 355 2 2 M 356 (4, 2, 1) 297 2 M 356 (4, 0, 2) 306 2 2 M 356 (4, 2, 2) 286 2 2 M 356 (4, 0, 0 1 ) 250 S 3 M 356 (2, 1, 0 1 ) 257 2 2 M 356 (0, 1, 0 2 ) 341 2 2 M 356 (2, 0, 0 1 ) 258 2 2 M 356 (0, 0, 1) 348 3 M 356 (0, 0, 2) 304 S 3 M 356 (0, 2, 0 2 ) 280 D 8 M 357 (2, 2, 2) 383 2 M 357 (4, 2, 4 1 ) 387 2 M 357 (4, 2, 6) 392 2 M 357 (0, 2, 6) 370 2 M 357 (0, 6, 2) 360 S 3 M 358 (2, 6, 4 2 ) 390 2 M 358 (4, 2, 6) 391 2 M 358 (4, 6, 4 2 ) 388 3 M 358 (0, 2, 4 1 ) 369 2 M 358 (0, 2, 6) 368 2 M 358 (0, 4, 2) 367 2 M 358 (0, 2, 4 2 ) 363 2 2 M 358 (0, 6, 2) 359 S 3 *********** **** **** *********** **** **** M 366 (0, 0, 0 1 : 2) 271 2 M 366 (1, 2, 0 1 ) 298 2 M 366 (2, 1, 0 2 ) 266 D 8 M 366 (2, 2, 1) 277 A 4 M 367 (1, 6, 2 1 ) 331 2 M 367 (1, 6, 2 2 ) 319 2 M 367 (1, 2, 2 1 ) 330 2 M 367 (1, 2, 2 2 ) 320 2 M 367 (0, 2, 0) 338 2 M 367 (0, 6, 2 1 ) 322 4 the electronic journal of combinatorics 17 (2010), #N4 7 Γ Numb er G 123 Γ Numb er G 123 M 367 (2, 4, 0) 291 2 M 367 (2, 4, 2 1 ) 293 2 M 367 (2, 2, 1) 290 2 M 367 (2, 2, 0) 292 2 M 367 (2, 6, 2 1 ) 284 4 M 367 (2, 6, 2 2 ) 281 S 3 M 368 (0, 6, 1 1 ) 349 2 M 368 (1, 6, 2) 345 2 M 368 (2, 4, 1 2 ) 299 1 M 368 (2, 4, 0) 288 2 M 368 (2, 2, 1 1 ) 289 2 M 368 (2, 2, 1 2 ) 287 3 M 368 (2, 6, 2) 283 2 2 *********** **** **** *********** **** **** M 377 (2, 2, 8 2 ) 424 2 M 377 (4, 6, 4) 419 2 M 377 (6, 6, 8 2 ) 415 S 3 M 378 (6, 2, 6 1 ) 418 2 M 378 (2, 2, 4 : 2) 423 2 M 378 (2, 6, 6 2 ) 414 1 *********** **** **** *********** **** **** M 388 (2, 6, 5 2 ) 410 2 M 388 (6, 4, 7 1 ) 411 2 *********** **** **** *********** **** **** M 555 (0, 4 1 , 2) 356 2 M 555 (4 2 , 4 1 , 4 2 ) 378 2 M 556 (4 1 , 2, 0 2 ) 307 2 M 556 (4 1 , 2, 2) 308 2 M 556 (4 2 , 1, 2) 309 2 M 556 (2, 0 1 , 1) 259 2 M 556 (4 1 , 0 1 , 1) 251 3 M 556 (0, 2, 0 2 ) 300 2 3 M 556 (0, 0 1 , 0 1 ) 244 2 4 2 2 M 557 (0, 4 2 , 6) 354 2 M 557 (4 1 , 4 2 , 2) 379 1 M 558 (0, 4 1 , 4 1 ) 353 1 M 558 (4 2 , 4 2 , 4 2 ) 377 1 M 558 (4 1 , 6, 2) 395 2 M 558 (0, 4 2 , 4 2 ) 352 3 M 558 (4 2 , 2, 6) 376 2 2 *********** **** **** *********** **** **** M 566 (1, 0 2 , 0 1 ) 275 1 M 566 (0 1 , 1, 0 1 ) 260 2 M 566 (2, 0 2 , 0 1 ) 267 2 M 566 (2, 2, 1) 302 2 2 M 566 (0 1 , 0 2 , 0 1 ) 249 2 2 M 566 (2, 2, 0 1 ) 303 2 2 M 566 (0 1 , 0 1 , 0 2 ) 245 2 3 2 2 M 566 (0 1 , 2, 0 2 ) 246 2 3 2 M 567 (0 1 , 6, 0) 248 2 2 M 567 (0 1 , 4 1 , 0) 247 D 8 M 567 (0 2 , 4 1 , 0) 334 2 M 567 (0 2 , 4 1 , 2 1 ) 256 2 M 567 (1, 2, 0) 340 1 M 567 (1, 4 1 , 0) 339 1 M 567 (1, 2, 2 1 ) 324 2 M 567 (0 1 , 4 2 , 1) 262 1 M 567 (0 2 , 4 1 , 2 1 ) 323 2 M 567 (2, 4 2 , 2 1 ) 314 1 M 567 (2, 4 2 , 2 2 ) 305 2 M 567 (2, 4 1 , 0) 301 2 2 M 568 (0 1 , 4 1 , 1 1 ) 252 2 M 568 (0 1 , 4 2 , 1 1 ) 255 2 M 568 (0 1 , 4 1 , 2) 261 2 M 568 (0 1 , 6, 1 2 ) 253 2 2 M 568 (0 1 , 6, 0) 254 2 2 M 568 (2, 4 2 , 1 1 ) 312 1 M 568 (2, 4 2 , 2) 311 2 M 568 (2, 6, 1 1 ) 313 2 M 568 (2, 6, 1 2 ) 310 2 *********** **** **** *********** **** **** M 577 (4 1 , 4 2 , 4) 386 1 M 577 (4 1 , 4 1 , 8 1 ) 384 2 M 577 (6, 2, 8 1 ) 382 2 M 577 (4 1 , 6, 4) 385 2 M 577 (2, 4 2 , 4) 380 2 M 577 (2, 4 1 , 8 1 ) 381 2 the electronic journal of combinatorics 17 (2010), #N4 8 Γ Number G 123 Γ Number G 123 M 588 (4 2 , 6, 8) 389 2 M 588 (2, 2, 7 1 ) 397 2 M 588 (2, 2, 5 2 ) 396 2 *********** **** **** *********** **** **** M 666 (0 1 , 0 2 , 0 2 ) 268 2 2 M 666 (0 1 , 0 2 , 1) 272 2 M 667 (0 2 , 1, 0) 263 2 2 M 667 (0 2 , 2 1 , 2 1 ) 264 4 M 667 (0 2 , 0, 1) 274 1 M 667 (1, 2 1 , 2 1 ) 315 2 M 667 (1, 0, 0) 316 1 M 667 (1, 2 1 , 2 2 ) 317 2 M 667 (0 1 , 2 1 , 2 1 ) 332 1 M 668 (0 2 , 1 2 , 1 2 ) 265 2 M 668 (0 2 , 1 1 , 1 2 ) 269 2 M 668 (0 2 , 2, 2) 270 2 M 668 (0 2 , 2, 1) 273 2 *********** **** **** *********** **** **** M 677 (2 2 , 1, 4) 318 1 M 677 (2 1 , 2 1 , 8 2 ) 321 2 M 677 (2 1 , 0, 4) 328 1 M 677 (2 1 , 0, 8 1 ) 329 1 M 677 (0, 1, 4) 337 1 M 678 (2 1 , 1 2 , 6 1 ) 325 1 M 678 (2 1 , 0, 6 1 ) 326 2 M 678 (2 1 , 0, 8) 327 2 M 678 (2 1 , 1 2 , 8) 333 1 M 678 (0, 1 1 , 6 1 ) 335 1 M 678 (0, 0, 4) 336 2 *********** **** **** *********** **** **** M 688 (1 2 , 0, 3) 342 1 M 688 (1 2 , 2, 5 2 ) 343 1 M 688 (1 1 , 2, 7 2 ) 350 1 M 688 (2, 2, 7 2 ) 351 2 *********** **** **** *********** **** **** M 777 (8 2 , 4, 4) 425 2 2 M 778 (4, 6 1 , 6 1 ) 426 1 *********** **** **** *********** **** **** M 888 (7 2 , 3, 7 2 ) 427 3 M 888 (7 2 , 5, 5) 428 1 M 888 (7 2 , 3, 3) 429 1 M 888 (3, 5 2 , 5 2 ) 430 1 M 888 (3, 7 1 , 7 1 ) 431 1 References [1] Buekenhout, Francis, editor,Handbook of Incidence Geometry, Buildings and Foun- dations. Elsevier, Amsterdam,1995. [2] J.J. Cannon and C. Playoust. An Introduction to Algebraic Programming with Magma [draft], Springer-Verlag (1997) . [3] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, 1985. [4] Curtis, R. T., A new combinatorial approach to M 24 . Math. Proc. Camb. Phil. Soc. 79(1976), 25-42. [5] Kilic, Nayil; Rowley, Peter, On rank 2 and rank 3 residually connected geometries for M 22 . Note di Matematica, 22(2003), 107-154. [6] Leemans, Dimitri, (2008) http://maths.ulb.ac.be/ dleemans/abstracts/m22pri.html the electronic journal of combinatorics 17 (2010), #N4 9 . rank 3 residually connected geometries for M 22 . Note di Matematica, 22(2003), 107–154. 1 Introduction Here we report on calculations carried out using Magma[2] on certain rank 3 geometries for. 2010 Mathematics Subject Classifications: 20D08, 51E10, 05C25 Abstract The rank 3 residually connected flag transitive geometries Γ for M 22 for which the stabilizer of each object in Γ is a maximal subgroup. Rank three residually connected geometries for M 22 , revisited Dimitri Leemans Universite Libre de Bruxelles Departement