Data to accompany the paper "The Strongly Regular (40,12,2,4) Graphs" author: E. Spence Department of Mathematics University of Glasgow Glasgow G12 8QQ Scotland The Electronic Journal of Combinatorics, Vol 7, 2000, R22. http://www.combinatorics.org _________________________________________________________________________ The following is a list of the adjacency matrices of the 28 strongly regular (40,12,2,4) graphs. Included are data concerning the automorphism group, the orbits and the distribution of the neighbour graphs among the five possible types: 011000000000 011000000000 011000000000 011000000000 011000000000 100100000000 101000000000 100100000000 101000000000 101000000000 100010000000 110000000000 100010000000 110000000000 110000000000 010001000000 000011000000 010001000000 000011000000 000011000000 001000100000 000100100000 001001000000 000101000000 000101000000 000100010000 000100010000 000110000000 000110000000 000110000000 000010001000 000010001000 000000011000 000000011000 000000011000 000001000100 000001000100 000000100100 000000100100 000000101000 000000100010 000000100010 000000100010 000000100010 000000110000 000000010001 000000010001 000000010001 000000010001 000000000011 000000001001 000000001001 000000001001 000000001001 000000000101 000000000110 000000000110 000000000110 000000000110 000000000110 I II III IV V This data is in the form of a five-tuple (a, b, c, d, e), where a denotes the number of neighbour graphs of type I, b the number of type II, ..., and e the number of type V. If the zeros on the diagonal of the adjacency matrices are replaced by ones the resulting matrix is the incidence matrix of a 2-(40,13,4) design (with a polarity having 40 absolute points). These designs, which are all non-isomorphic, were investigated for further polarities, and where these exist the numbers of absolute points are given. __________________________________________________________________________ Number 1. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1100100000000100000000111111110000000000 1100010000000010000000110000001111110000 1010001000000001100000001000001110001110 1001001000000000011000000111000001001101 1000110000000000000111000000110000110011 1000000011000110000100100100001000001011 1000000100100001010010010100100100100100 1000000100010000101001010010011000010100 1000000010001100100010001010000011010001 1000000001001010010001001001100101000010 1000000000110001001100100001010010101000 0110000100100000000001000001000001111110 0101000100010000000010001000110010001101 0100100010001000010001010000011001001001 0100100001100000001100100000100101000101 0100010010010001000100101010000000010110 0100010001001000100010011100000000101010 0100001100001000110000000101101010010000 0100001010100010001000000011011100000010 0100001001010101000000000110000110100001 0011000100001000110000000010010100100011 0011000011000001001000000001100010010011 0010100000110010011000000100001000110001 0010010110000000001101001000010111000000 0010010001100000010011100000101010001000 0010010000011100000110010000001100000101 0010001010010010100100010010000001101000 0010001001001011000010100100000001010100 0001100101000001000110001011000001100000 0001100010010000100011100101000000011000 0001100000101010000101010110000000000110 0001010000110101100000000100111000000010 0001001010001100001001101000101000000100 0001001001100100010100011000010100001000 0000110100001111001000000010100100010000 0000110011000110110000000001010010100000 0000101100010100011010110000000011000000 0000011100100011100001111001000000000000 Order of automorphism group: 48 Generators of the automorphism group 1. (3,4)(5,6)(9,10)(11,12)(14,15)(16,19)(17,18)(21,22)(25,34)(26,31) (27,32)(28,33)(29,36)(30,35)(39,40) 2. (2,3)(4,5)(6,7)(8,11)(10,13)(15,25)(16,24)(17,23)(18,29)(19,30)(20,27) (21,26)(22,28)(31,33)(34,39)(35,38)(36,37) 3. (1,20)(3,17)(4,18)(5,29)(6,36)(8,13)(9,33)(10,28)(11,26)(12,31)(14,19) (15,16)(21,22)(24,38)(25,34)(27,32)(30,40)(35,39) Orbits under the action of the full automorphism group (1,20,27,32)(2,3,4,5,6,7,17,18,23,29,36,37)(8,9,10,11,12,13,21,22,26,28,31,33)(14,15,16,19,24,25,30,34,35,38,39,40) Distribution of neighbour graphs: (12, 0, 28, 0, 0) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 2. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000100100100010010010010010010010010 1000000100010010010001010100001001001001 1000000010001010001100001001001100001010 1000000001001001001010001010100010100001 1000000000110001100001100001010001010100 0100100110000000001001001011000001000101 0100100001100000100010100101000010000011 0100100000011000010100010110000100000110 0100010100001010000010000010110101001000 0100010011000001000001000001101110010000 0100010000110100000100000100011011100000 0100001100100001001000110000100000011001 0100001010010010100000011000001000110100 0100001001001100010000101000010000101010 0010100100001010000101000010001010110000 0010100011000001000110000001010001101000 0010100000110100000011000100100100011000 0010010101000011001000001000010000010110 0010010010010101100000100000001000001011 0010010000101110010000010000100000100101 0010001100010000110100001001000011000010 0010001010001000101001010100000110000001 0010001001100000011010100010000101000100 0001100100100001110000001000011000100001 0001100010010010011000100000110000001100 0001100001001100101000010000101000010010 0001010100010000001011110001000100000010 0001010010001000010110101100000001000001 0001010001100000100101011010000010000100 0001001100001101000010000101001010001000 0001001010100011000001000110100001100000 0001001001010110000100000011010100010000 Order of automorphism group: 384 Generators of the automorphism group 1. (8,13)(9,12)(10,11)(14,16)(18,19)(20,22)(24,25)(26,28)(29,30)(32,34) (35,36)(39,40) 2. (3,4)(6,7)(8,9)(10,11)(12,13)(17,21)(18,20)(19,22)(23,33)(24,32)(25,34) (26,39)(27,38)(28,40)(29,36)(30,35)(31,37) 3. (2,3)(5,6)(9,10)(11,12)(14,26)(15,27)(16,28)(17,23)(18,24)(19,25) (20,29)(21,31)(22,30)(32,35)(33,37)(34,36)(39,40) 4. (2,5)(3,6)(4,7)(17,23)(18,24)(19,25)(20,32)(21,33)(22,34)(29,35)(30,36) (31,37) 5. (1,2)(5,15)(6,21)(7,17)(8,20)(9,19)(10,16)(11,14)(12,18)(13,22)(24,26) (25,28)(27,31)(32,40)(34,39)(35,36)(37,38) Orbits under the action of the full automorphism group (1,2,3,4,5,6,7,15,17,21,23,27,31,33,37,38)(8,9,10,11,12,13,14,16,18,19,20,22,24,25,26,28,29,30,32,34,35,36,39,40) Distribution of neighbour graphs: ( 0, 0, 24, 16, 0) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 3. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000100100100010010010010010010010010 1000000100010010010001010100001001001001 1000000010001010001100001001001100001010 1000000001001001001010001010100010100001 1000000000110001100001100001010001010100 0100100110000000001001001011000001000101 0100100001100000110000101000010000100011 0100100000011000010100010110000100000110 0100010100001010000010000010110101001000 0100010011000011000000000001101010010100 0100010000110100000100000100011011100000 0100001100100001001000110000100000011001 0100001010010000100001011000001100110000 0100001001001100000010100101000010001010 0010100100001010000101000010001010110000 0010100011000001000110000001010001101000 0010100000110110000010000100100000011100 0010010101000001001001001000010100010010 0010010010010101100000100000001000001011 0010010000101100010001010000100100100001 0010001100010000110100001001000011000010 0010001010001010101000010100000010000101 0010001001100000011010100010000101000100 0001100100100001100010000101001010000001 0001100010010000011001100000110100001000 0001100001001100101000010000101000010010 0001010100010010001010110001000000000110 0001010010001000010110101100000001000001 0001010001100000100101011010000010000100 0001001100001101010000001000011000101000 0001001010100011000001000110100001100000 0001001001010110000100000011010100010000 Order of automorphism group: 8 Generators of the automorphism group 1. (2,4)(5,7)(8,10)(9,12)(11,13)(14,40)(15,38)(16,39)(17,37)(18,35)(19,36) (20,34)(21,33)(22,32)(23,31)(24,29)(25,30) 2. (2,5)(3,6)(4,7)(8,13)(9,12)(10,11)(14,16)(17,23)(18,25)(19,24)(20,34) (21,33)(22,32)(26,28)(29,36)(30,35)(31,37)(39,40) 3. (1,27)(2,17)(3,6)(4,37)(5,23)(7,31)(8,14)(10,40)(11,39)(13,16)(18,30) (19,29)(20,34)(22,32)(24,36)(25,35)(26,28) Orbits under the action of the full automorphism group (1,27)(2,4,5,7,17,23,31,37)(3,6)(8,10,11,13,14,16,39,40)(9,12)(15,38)(18,25,30,35)(19,24,29,36)(20,34)(21,33)(22,32)(26,28) Distribution of neighbour graphs: (10, 8, 18, 4, 0) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 4. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000100100100010010010010010010010010 1000000100010010010001010100001001001001 1000000010001010001100001001001100001010 1000000001001001001010001010100010100001 1000000000110001100001100001010001010100 0100100110000001001000000001101001010001 0100100001100000100010100101000010000011 0100100000011100000100010100010000101010 0100010100001010000010000010110101001000 0100010011000000000101001001010011100000 0100010000110100000001000110001110000100 0100001100100001010000101010000000001101 0100001010010010100000011000001000110100 0100001001001000011000110000100100010010 0010100100001010000101000010001010110000 0010100011000001000011000011000100001100 0010100000110000010110000100100101010000 0010010101000011001000001000010000010110 0010010010010000101100110000000001000011 0010010000101110010000010000100000100101 0010001100010100100001001001000010001010 0010001010001001110000000100001110000001 0010001001100100001010100000010001101000 0001100100100000101001011000010000100001 0001100010010010011000100000110000001100 0001100001001100110000001010001000000110 0001010100010001010010100001001100000010 0001010010001100000011101100000000001001 0001010001100001100100010000101010010000 0001001100001000001110010101000011000000 0001001010100011000001000110100001100000 0001001001010110000100000011010100010000 Order of automorphism group: 12 Generators of the automorphism group 1. (3,4)(6,7)(8,12)(9,13)(14,16)(17,21)(18,22)(19,20)(23,33)(24,34)(25,32) (26,40)(27,38)(28,39)(29,35)(30,36)(31,37) 2. (1,6,7)(2,18,22)(3,28,29)(4,35,39)(8,26,38)(9,36,21)(10,19,20)(11,37,31) (12,27,40)(13,17,30)(14,25,24)(15,33,23)(16,34,32) 3. (1,15)(3,17)(4,21)(6,23)(7,33)(8,26)(9,39)(12,40)(13,28)(14,16)(18,22) (19,20)(24,34)(25,32)(29,30)(31,37)(35,36) Orbits under the action of the full automorphism group (1,6,7,15,23,33)(2,18,22)(3,4,9,13,17,21,28,29,30,35,36,39)(8,12,26,27,38,40)(10,19,20)(11,31,37)(14,16,24,25,32,34) Distribution of neighbour graphs: ( 6, 12, 12, 10, 0) The 2-(40,13,4) design has polarities with 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 5. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000100100100010010010010010010010010 1000000100010010100010010001001001001001 1000000010001010001001001001100100010001 1000000001001001010001001100010010001100 1000000000110001001100100010001001100010 0100100110000000001001001000011001101000 0100100001100000010100100000101010011000 0100100000011000100010010000110100110000 0100010101000001000001101010000000010011 0100010010010010000100011001000000100110 0100010000101100000010110100000000001101 0100001100001010010000000110010101000001 0100001011000001001000000101001110000010 0100001000110100100000000011100011000100 0010100100001010101000000001010010000110 0010100011000001011000000010100001000101 0010100000110100110000000100001100000011 0010010100010000001110001000100011010000 0010010010001000100101010000001110001000 0010010001100000010011100000010101100000 0010001100100011000001010100000000101010 0010001010010101000100100001000000011001 0010001001001110000010001010000000110100 0001100100100001000110001011000000001100 0001100010010010000011100110000000100001 0001100001001100000101010101000000010010 0001010100001101010000000001101010000001 0001010010100011100000000100011001000100 0001010001010110001000000010110100000010 0001001100010000011001110000001100010000 0001001010001000110010101000100001001000 0001001001100000101100011000010010100000 Order of automorphism group: 64 Generators of the automorphism group 1. (8,13)(9,12)(10,11)(14,16)(17,19)(21,22)(24,25)(26,27)(29,31)(32,34) (36,37)(38,39) 2. (2,6)(3,7)(4,5)(8,10)(9,12)(11,13)(14,37)(15,35)(16,36)(20,28)(21,26) (22,27)(23,40)(24,38)(25,39)(29,31)(32,34) 3. (1,2,4,3)(5,15,33,23)(6,20,35,30)(7,18,40,28)(8,14,37,27,13,16,36,26) (9,19,32,31,12,17,34,29)(10,22,39,24,11,21,38,25) Orbits under the action of the full automorphism group (1,2,3,4,5,6,7,15,18,20,23,28,30,33,35,40)(8,10,11,13,14,16,21,22,24,25,26,27,36,37,38,39)(9,12,17,19,29,31,32,34) Distribution of neighbour graphs: (16, 0, 8, 16, 0) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 6. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100000001010001101000010000110 0100100010010000100001100110000001000011 0100100001001000010100010011000100000101 0100010100001010000001000001101110010000 0100010010100001000100000100110011001000 0100010001010100000010000010011101100000 0100001100010001010000101000001000110010 0100001010001100001000110000100000011001 0100001001100010100000011000010000101100 0010100100001010000110000001010001101000 0010100010100001000011000100001100110000 0010100001010100000101000010100010011000 0010010100010110010000010000001000001101 0010010010001011001000001000100000100110 0010010001100101100000100000010000010011 0010001100100000110010001010000101000001 0010001010010000011001100001000110000100 0010001001001000101100010100000011000010 0001100100010001101000010000110000001010 0001100010001100110000001000011000100001 0001100001100010011000100000101000010100 0001010100100000001101110010000010000001 0001010010010000100110011001000001000100 0001010001001000010011101100000100000010 0001001100001101000001000110010001010000 0001001010100110000100000011001100001000 0001001001010011000010000101100010100000 Order of automorphism group: 51840 Generators of the automorphism group 1. (8,9,10)(11,12,13)(14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28) (29,30,31)(32,33,34)(35,36,37)(38,39,40) 2. (8,11)(9,13)(10,12)(15,16)(17,18)(20,22)(23,24)(26,28)(30,31)(32,34) (36,37)(38,39) 3. (5,6,7)(11,13,12)(14,17,20)(15,18,21)(16,19,22)(23,26,29)(24,27,30) (25,28,31)(32,35,38)(33,36,39)(34,37,40) 4. (5,8)(6,10)(7,9)(15,20)(16,17)(18,22)(24,29)(25,26)(27,31)(33,38) (34,35)(36,40) 5. (3,4)(6,7)(9,10)(12,13)(15,16)(17,20)(18,22)(19,21)(23,32)(24,34) (25,33)(26,38)(27,40)(28,39)(29,35)(30,37)(31,36) 6. (2,3)(6,7)(9,10)(11,13)(14,23)(15,25)(16,24)(17,29)(18,31)(19,30) (20,26)(21,28)(22,27)(33,34)(35,38)(36,40)(37,39) 7. (2,5)(3,6)(4,7)(17,23)(18,24)(19,25)(20,32)(21,33)(22,34)(29,35)(30,36) (31,37) 8. (1,2)(5,14)(6,21)(7,19)(8,15)(9,17)(10,22)(11,16)(12,20)(13,18)(23,26) (24,28)(27,29)(30,31)(32,39)(34,38)(35,40) Graph is transitive Distribution of neighbour graphs: ( 0, 0, 0, 0, 40) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 7. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100010001000000101001010000110 0100100010010100000001100010100000011001 0100100001001000010100010011000100000101 0100010100001000000011001001100110010000 0100010010100001000100000100110011001000 0100010001010100000010000010011101100000 0100001100010001010000101000001000110010 0100001010001000101000110100000001000011 0100001001100010100000011000010000101100 0010100100001010000110000001010001101000 0010100010100001000011000100001100110000 0010100001010000100101000110000011000010 0010010100010100010010011000000000001101 0010010010001011001000001000100000100110 0010010001100101100000100000010000010011 0010001100100010110000000010001101000001 0010001010010000011001100001000110000100 0010001001001100001100010000100010011000 0001100100010001101000010000110000001010 0001100010001100110000001000011000100001 0001100001100000011010101000100000010100 0001010100100000001101110010000010000001 0001010010010010100100010001001001000100 0001010001001010010001100100001100000010 0001001100001101000001000110010001010000 0001001010100100000110001011000100001000 0001001001010011000010000101100010100000 Order of automorphism group: 192 Generators of the automorphism group 1. (5,13)(6,11)(7,12)(14,17)(15,21)(19,22)(24,27)(25,31)(26,29)(32,38) (34,37)(36,39) 2. (2,3)(5,7)(9,10)(12,13)(14,29)(15,31)(16,30)(17,26)(18,28)(19,27) (20,23)(21,25)(22,24)(32,38)(33,40)(34,39)(36,37) 3. (2,5)(3,7)(4,6)(8,11)(9,12)(10,13)(17,34)(18,32)(19,33)(20,24)(21,25) (22,23)(26,39)(27,40)(28,38) 4. (1,4)(5,34)(6,38)(7,36)(8,35)(9,40)(10,33)(11,32)(12,39)(13,37)(14,19) (15,21)(16,18)(17,22)(24,29)(26,27)(28,30) Orbits under the action of the full automorphism group (1,4,6,8,11,16,18,28,30,32,35,38)(2,3,5,7,9,10,12,13,14,17,19,20,22,23,24,26,27,29,33,34,36,37,39,40)(15,21,25,31) Distribution of neighbour graphs: ( 0, 0, 4, 24, 12) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 8. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100010001000000101001010000110 0100100010010100000001100010100000011001 0100100001001000100100010011000100000011 0100010100001001000001000100110011010000 0100010010100000000110001001100110001000 0100010001010100000010000010011101100000 0100001100010001010000101000001000110010 0100001010001000011000110100000001000101 0100001001100010100000011000010000101100 0010100100001010000110000001010001101000 0010100010100001000011000100001100110000 0010100001010000010101000110000011000100 0010010100010100100010011000000000001011 0010010010001011001000001000100000100110 0010010001100101010000100000010000010101 0010001100100010110000000010001101000001 0010001010010000101001100001000110000010 0010001001001100001100010000100010011000 0001100100010001011000010000110000001100 0001100010001100110000001000011000100001 0001100001100000101010101000100000010010 0001010100100000001101110010000010000001 0001010010010010100100010001001001000100 0001010001001010010001100100001100000010 0001001100001100000011001011000100010000 0001001010100101000100000110010001001000 0001001001010011000010000101100010100000 Order of automorphism group: 8 Generators of the automorphism group 1. (2,7)(3,5)(4,6)(8,11)(9,13)(10,12)(14,29)(15,31)(16,30)(17,39)(18,38) (19,40)(20,22)(23,24)(26,34)(27,33)(28,32)(36,37) 2. (2,9)(3,10)(4,8)(5,12)(6,11)(7,13)(14,36)(16,30)(17,39)(19,24)(20,33) (22,27)(23,40)(26,34)(29,37) 3. (1,35)(2,20)(3,23)(4,8)(5,24)(6,11)(7,22)(9,33)(10,40)(12,19)(13,27) (14,36)(15,31)(17,39)(21,25)(26,34)(29,37) Orbits under the action of the full automorphism group (1,35)(2,7,9,13,20,22,27,33)(3,5,10,12,19,23,24,40)(4,6,8,11)(14,29,36,37)(15,31)(16,30)(17,39)(18,38)(21,25)(26,34)(28,32) Distribution of neighbour graphs: ( 2, 8, 12, 16, 2) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 9. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100010001000000101001010000110 0100100010010100000001100010100000011001 0100100001001000100100010110000001000011 0100010100001001000001000001110110010000 0100010010100000000110001100100011001000 0100010001010100000010000010011101100000 0100001100010001010000101000001000110010 0100001010001000011000110001000100000101 0100001001100010100000011000010000101100 0010100100001010000110000001010001101000 0010100010100001000011000100001100110000 0010100001010000010101000011000110000100 0010010100010101010000010000010000001101 0010010010001011001000001000100000100110 0010010001100100100010101000000000010011 0010001100100010110000000010001101000001 0010001010010000101001100100000011000010 0010001001001100001100010000100010011000 0001100100010000101010011000100000001010 0001100010001100110000001000011000100001 0001100001100001011000100000110000010100 0001010100100000001101110010000010000001 0001010010010010100100010001001001000100 0001010001001010010001100100001100000010 0001001100001100000011001110000001010000 0001001010100101000100000011010100001000 0001001001010011000010000101100010100000 Order of automorphism group: 48 Generators of the automorphism group 1. (5,9,13)(6,8,11)(7,10,12)(14,18,17)(15,21,16)(19,20,22)(23,24,27) (25,30,31)(26,29,28)(32,39,37)(34,36,38) 2. (2,3)(5,7)(9,10)(12,13)(14,29)(15,31)(16,30)(17,26)(18,28)(19,27) (20,23)(21,25)(22,24)(32,38)(33,40)(34,39)(36,37) 3. (2,5)(3,7)(4,6)(8,11)(9,12)(10,13)(17,34)(18,32)(19,33)(20,24)(21,25) (22,23)(26,39)(27,40)(28,38) 4. (1,35)(2,19,10,40,7,20)(3,27,9,33,5,23)(4,6,8)(12,22)(13,24)(14,34,28,29,39,18) (15,30,25)(16,21,31)(17,32,37,26,38,36) Orbits under the action of the full automorphism group (1,35)(2,3,5,7,9,10,12,13,19,20,22,23,24,27,33,40)(4,6,8,11)(14,17,18,26,28,29,32,34,36,37,38,39)(15,16,21,25,30,31) Distribution of neighbour graphs: ( 0, 16, 6, 12, 6) The 2-(40,13,4) design has polarities with 4, 10, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 10. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100010100000001001010000001110 0100100010010100000010000110001001100001 0100100001001000010100010011000100000101 0100010100001100001000010000101010010001 0100010010100001000100000100110011001000 0100010001010000100001100010010101000010 0100001100010001010000101000001000110010 0100001010001010000001100001100100011000 0100001001100000001010011100000010100100 0010100100001000001110010000010001101000 0010100010100001100001100100000000010011 0010100001010100000101000010100010011000 0010010100010010010001010000001100001100 0010010010001011001000001000100000100110 0010010001100101000010000000011100110000 0010001100100000110010001010000101000001 0010001010010100011000100001000010000101 0010001001001010100100000101000011000010 0001100100010001001010000101100010000010 0001100010001000110001001000011100100000 0001100001100010011000100000101000010100 0001010100100010000101100011000010000001 0001010010010000100110011001000001000100 0001010001001100010010101100000000000011 0001001100001101000001000110010001010000 0001001010100100001100010010001100001000 0001001001010011100000010000110000101000 Order of automorphism group: 16 Generators of the automorphism group 1. (5,10)(6,9)(7,8)(14,22)(15,19)(17,21)(23,31)(24,28)(26,30)(32,40) (33,37)(35,39) 2. (3,4)(6,7)(8,9)(11,12)(14,15)(17,21)(18,20)(19,22)(23,33)(24,32)(25,34) (26,39)(27,38)(28,40)(29,36)(30,35)(31,37) 3. (1,13)(2,16)(3,27)(4,38)(5,17,10,21)(6,31,9,23)(7,33,8,37)(11,12) (14,40,22,32)(15,24,19,28)(18,20)(25,29)(26,39,30,35)(34,36) 4. (1,18,13,20)(2,11,16,12)(3,29,27,25)(4,34,38,36)(5,26,21,35,10,30,17,39) (6,37,23,8,9,33,31,7)(14,24,32,15,22,28,40,19) Orbits under the action of the full automorphism group (1,13,18,20)(2,11,12,16)(3,4,25,27,29,34,36,38)(5,10,17,21,26,30,35,39)(6,7,8,9,23,31,33,37)(14,15,19,22,24,28,32,40) Distribution of neighbour graphs: ( 8, 8, 0, 20, 4) The 2-(40,13,4) design has polarities with 4, 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 11. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000100100100100100100100100100 1000000101000010010010010010010010010010 1000000110000001001001001001001001001001 1000000000011100010001010001100001100010 1000000000101010001100001100010100010001 1000000000110001100010100010001010001100 0100100100100010100000001001010000001110 0100100010010100000010000110001001100001 0100100001001000010100010011000100000101 0100010100001100001000010000101010010001 0100010010100001000100000100110011001000 0100010001010000100001100010010101000010 0100001100010001010000101000001000110010 0100001010001010000001100001100100011000 0100001001100000001010011100000010100100 0010100100001000001110010100000001001010 0010100010100001100001100000010000110001 0010100001010100000101000010100010011000 0010010100010010010001100001000010000101 0010010010001011001000001000100000100110 0010010001100101000010000100001100010010 0010001100100000110010001010000101000001 0010001010010100011000010000001100001100 0010001001001010100100000001010011100000 0001100100010001001010000001110010100000 0001100010001000110001001100001100000010 0001100001100010011000100000101000010100 0001010100100010000101010010001100001000 0001010010010000100110011001000001000100 0001010001001100010010101000010000100001 0001001100001101000001000110010001010000 0001001010100100001100100011000010000001 0001001001010011100000010100100000001010 Order of automorphism group: 144 Generators of the automorphism group 1. (5,6,7)(8,10,9)(11,12,13)(14,19,21)(15,17,22)(16,18,20)(23,28,30) (24,26,31)(25,27,29)(32,37,39)(33,35,40)(34,36,38) 2. (5,8,6,10,7,9)(11,13,12)(14,17,19,22,21,15)(16,20,18)(23,26,28,31,30,24) (25,29,27)(32,35,37,40,39,33)(34,38,36) 3. (3,4)(6,7)(8,9)(11,12)(14,15)(17,21)(18,20)(19,22)(23,33)(24,32)(25,34) (26,39)(27,38)(28,40)(29,36)(30,35)(31,37) 4. (2,3)(6,7)(8,9)(12,13)(14,24)(15,23)(16,25)(17,30)(18,29)(19,31)(20,27) (21,26)(22,28)(32,33)(35,39)(36,38)(37,40) 5. (1,11,13)(2,18,16)(3,29,27)(4,34,38)(5,14,37,10,22,33)(6,28,21,9,24,17) (7,39,23,8,35,31)(15,30,40,19,26,32) Orbits under the action of the full automorphism group (1,11,12,13)(2,3,4,16,18,20,25,27,29,34,36,38)(5,6,7,8,9,10,14,15,17,19,21,22,23,24,26,28,30,31,32,33,35,37,39,40) Distribution of neighbour graphs: ( 0, 24, 0, 12, 4) The 2-(40,13,4) design has polarities with 16 and 22 abs. pts. -------------------------------------------------------------------------- Number 12. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000010100010001001010001 0100010010010010000001010001100100001100 0100010010010001000100001100010010100010 0100001001001100001000010010001010001100 0100001001001010100000001001100001100010 0100001001001001010000100100010100010001 0010100001010100100001000001010010001010 0010100001010010010100000100001001100001 0010100001010001001010000010100100010100 0010010100001100001001010000100001010010 0010010100001010100100001000010100001001 0010010100001001010010100000001010100100 0010001010100100010010001100000010001001 0010001010100010001001100010000001100100 0010001010100001100100010001000100010010 0001100010001100010001001010001000100010 0001100010001010001100100001100000010001 0001100010001001100010010100010000001100 0001010001100100001010010001010100000001 0001010001100010100001001100001010000100 0001010001100001010100100010100001000010 0001001100010100010100001001010001010000 0001001100010010001010100100001100001000 0001001100010001100001010010100010100000 Order of automorphism group: 27 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (2,3,4)(8,10,9)(11,12,13)(14,23,33,15,24,34,16,25,32)(17,28,36,18,26,37,19,27,35) (20,30,40,21,31,38,22,29,39) 3. (2,8,12)(3,10,13)(4,9,11)(5,7,6)(14,38,18,15,39,19,16,40,17)(20,28,24,21,26,25,22,27,23) (29,36,32,30,37,33,31,35,34) Orbits under the action of the full automorphism group (2,3,4,8,9,10,11,12,13)(5,6,7)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40) Distribution of neighbour graphs: (27, 12, 0, 0, 1) The 2-(40,13,4) design has no further polarities. -------------------------------------------------------------------------- Number 13. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000010100010001001010001 0100010010010010000001010001100100001100 0100010010010001000100001100010010100010 0100001001001100001000010010100001001010 0100001001001010100000001001010100100001 0100001001001001010000100100001010010100 0010100001010100100001000001010010001010 0010100001010010010100000100001001100001 0010100001010001001010000010100100010100 0010010100001100001001010000010100010001 0010010100001010100100001000001010001100 0010010100001001010010100000100001100010 0010001010100100010100001001000010010001 0010001010100010001010100100000001001100 0010001010100001100001010010000100100010 0001100010001100010010001100001000001010 0001100010001010001001100010100000100001 0001100010001001100100010001010000010100 0001010001100100001010010001001010000100 0001010001100010100001001100100001000010 0001010001100001010100100010010100000001 0001001100010100010001001010010001100000 0001001100010010001100100001001100010000 0001001100010001100010010100100010001000 Order of automorphism group: 9 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (2,9,13)(3,8,11)(4,10,12)(5,7,6)(14,29,28,16,31,27,15,30,26)(17,33,21,19,32,20,18,34,22) (23,40,35,25,39,37,24,38,36) Orbits under the action of the full automorphism group (2,9,13)(3,8,11)(4,10,12)(5,6,7)(14,15,16,26,27,28,29,30,31)(17,18,19,20,21,22,32,33,34)(23,24,25,35,36,37,38,39,40) Distribution of neighbour graphs: (18, 12, 9, 0, 1) The 2-(40,13,4) design has no further polarities. -------------------------------------------------------------------------- Number 14. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000001001100100001100010 0100010010010001000010100001010100010100 0100001001001100100000001001010001010010 0100001001001010001000010100001100100001 0100001001001001010000100010100010001100 0010100001010100001001000100001010010010 0010100001010010100010000001100001001100 0010100001010001010100000010010100100001 0010010100001100010010100000010001001001 0010010100001010100001001000001100010100 0010010100001001001100010000100010100010 0010001010100100010001010001000100010001 0010001010100010001100001100000010001100 0010001010100001100010100010000001100010 0001100010001100001010001010100000001010 0001100010001010100001100001010000100001 0001100010001001010100010100001000010100 0001010001100100010010001001001010000100 0001010001100010001100100010100001000001 0001010001100001100001010100010100000010 0001001100010100001001010010010001100000 0001001100010010010100100001001100001000 0001001100010001100010001100100010010000 Order of automorphism group: 18 Generators of the automorphism group 1. (5,6,7)(8,9,10)(11,12,13)(14,17,20)(15,19,22)(16,18,21)(23,27,30) (24,28,29)(25,26,31)(32,37,39)(33,36,38)(34,35,40) 2. (3,4)(5,11)(6,12)(7,13)(15,16)(18,19)(21,22)(23,35)(24,37)(25,36) (26,38)(27,40)(28,39)(29,32)(30,34)(31,33) 3. (2,3)(5,8,6,9,7,10)(11,13,12)(14,28,17,29,20,24)(15,26,19,31,22,25) (16,27,18,30,21,23)(32,39,37)(33,40,36,34,38,35) Orbits under the action of the full automorphism group (2,3,4)(5,6,7,8,9,10,11,12,13)(14,17,20,24,28,29,32,37,39)(15,16,18,19,21,22,23,25,26,27,30,31,33,34,35,36,38,40) Distribution of neighbour graphs: (18, 18, 0, 3, 1) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 15. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000001100001100001100010 0100010010010001000010001100010100010100 0100001001001100100000001001010001010010 0100001001001010001000100010100100001001 0100001001001001010000010100001010100100 0010100001010100010010000100001001010001 0010100001010010100001000001010100001100 0010100001010001001100000010100010100010 0010010100001100001001100000010010001010 0010010100001010100010001000001001100100 0010010100001001010100010000100100010001 0010001010100100010010001001000010001100 0010001010100010001100010100000001100001 0010001010100001100001100010000100010010 0001100010001100001010010001001000100010 0001100010001010100001001100100000010001 0001100010001001010100100010010000001100 0001010001100100010001001010010100000001 0001010001100010001100100001001010000100 0001010001100001100010010100100001000010 0001001100010100001001010010100001010000 0001001100010010010100001100001100001000 0001001100010001100010100001010010100000 Order of automorphism group: 3 Generators of the automorphism group 1. (2,8,13)(3,10,11)(4,9,12)(5,6,7)(14,28,22)(15,27,21)(16,26,20)(17,40,33) (18,38,32)(19,39,34)(23,36,31)(24,35,29)(25,37,30) Orbits under the action of the full automorphism group (2,8,13)(3,10,11)(4,9,12)(5,6,7)(14,22,28)(15,21,27)(16,20,26)(17,33,40)(18,32,38)(19,34,39)(23,31,36)(24,29,35)(25,30,37) Distribution of neighbour graphs: (12, 18, 6, 3, 1) The 2-(40,13,4) design has no further polarities. -------------------------------------------------------------------------- Number 16. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000001100001100001100010 0100010010010001000010001100010100010100 0100001001001100100000001001010001010010 0100001001001010001000100100001010100001 0100001001001001010000010010100100001100 0010100001010100010010000001010100001001 0010100001010010100001000100001001010100 0010100001010001001100000010100010100010 0010010100001100001010010000100001001010 0010010100001010100001001000010100100001 0010010100001001010100100000001010010100 0010001010100100010001001100000010010001 0010001010100010001100100010000001001100 0010001010100001100010010001000100100010 0001100010001100001001100010001000010010 0001100010001010100010001001100000001100 0001100010001001010100010100010000100001 0001010001100100010010001010001001000100 0001010001100010001100010001100100000001 0001010001100001100001100100010010000010 0001001100010100001001010001010010100000 0001001100010010010100001100001100001000 0001001100010001100010100010100001010000 Order of automorphism group: 1 Distribution of neighbour graphs: (16, 14, 4, 5, 1) The 2-(40,13,4) design has no further polarities. -------------------------------------------------------------------------- Number 17. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000010001001100001100100 0100010010010001000001100100010100010010 0100001001001100100000001001010001010010 0100001001001010010000100100001100001001 0100001001001001001000010010100010100100 0010100001010100001010000001010010100001 0010100001010010100001000100001001010100 0010100001010001010100000010100100001010 0010010100001100001010010000100001001010 0010010100001010100001001000010100100001 0010010100001001010100100000001010010100 0010001010100100010001001100000010010001 0010001010100010001100100010000001001100 0010001010100001100010010001000100100010 0001100010001100001010001010001000010100 0001100010001010100001100001100000001010 0001100010001001010100010100010000100001 0001010001100100010001100010001001000010 0001010001100010001100010001100100000001 0001010001100001100010001100010010000100 0001001100010100010001010001010100001000 0001001100010010001100001100001010100000 0001001100010001100010100010100001010000 Order of automorphism group: 6 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (3,4)(5,11)(6,12)(7,13)(15,16)(18,19)(21,22)(23,35)(24,37)(25,36) (26,38)(27,40)(28,39)(29,32)(30,34)(31,33) Orbits under the action of the full automorphism group (3,4)(5,11)(6,12)(7,13)(14,15,16)(17,18,19)(20,21,22)(23,24,25,35,36,37)(26,27,28,38,39,40)(29,30,31,32,33,34) Distribution of neighbour graphs: (18, 20, 0, 0, 2) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 18. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000010001001100001100100 0100010010010001000001100100010100010010 0100001001001100100000001001010001010010 0100001001001010010000100100001100001001 0100001001001001001000010010100010100100 0010100001010100001010000010001001100010 0010100001010010100001000001100100010001 0010100001010001010100000100010010001100 0010010100001100001010001000100010010001 0010010100001010100001100000010001001100 0010010100001001010100010000001100100010 0010001010100100010001010100000001001010 0010001010100010001100001010000100100001 0010001010100001100010100001000010010100 0001100010001100001010010001010000001100 0001100010001010100001001100001000100010 0001100010001001010100100010100000010001 0001010001100100010001100001010010000001 0001010001100010001100010100001001000100 0001010001100001100010001010100100000010 0001001100010100010001001010001100010000 0001001100010010001100100001100010001000 0001001100010001100010010100010001100000 Order of automorphism group: 48 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (5,13)(6,12)(7,11)(8,10)(14,20)(15,22)(16,21)(18,19)(23,28)(24,27) (25,26)(29,30)(32,34)(35,39)(36,38)(37,40) 3. (3,4)(5,7)(8,10)(11,13)(14,20)(15,21)(16,22)(23,39)(24,40)(25,38) (26,36)(27,37)(28,35)(29,34)(30,32)(31,33) 4. (2,6)(3,5,4,7)(8,11,10,13)(9,12)(14,35,22,27,15,36,20,28,16,37,21,26) (17,18,19)(23,33,38,30,24,34,39,31,25,32,40,29) Orbits under the action of the full automorphism group (2,6,9,12)(3,4,5,7,8,10,11,13)(14,15,16,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(17,18,19) Distribution of neighbour graphs: ( 0, 32, 0, 0, 8) The 2-(40,13,4) design has polarities with 10 and 16 abs. pts. -------------------------------------------------------------------------- Number 19. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000010001001100001100100 0100010010010001000001100100010100010010 0100001001001100100000001001010001010010 0100001001001010010000100100001100001001 0100001001001001001000010010100010100100 0010100001010100001010000010100001001010 0010100001010010100001000001010100100001 0010100001010001010100000100001010010100 0010010100001100001010001000010010100001 0010010100001010100001100000001001010100 0010010100001001010100010000100100001010 0010001010100100010001100001000010010001 0010001010100010001100010100000001001100 0010001010100001100010001010000100100010 0001100010001100001010010001001000010100 0001100010001010100001001100100000001010 0001100010001001010100100010010000100001 0001010001100100010001010100001001000010 0001010001100010001100001010100100000001 0001010001100001100010100001010010000100 0001001100010100010001001010010100001000 0001001100010010001100100001001010100000 0001001100010001100010010100100001010000 Order of automorphism group: 9 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (2,10,12)(3,9,13)(4,8,11)(5,6,7)(14,35,39,15,36,40,16,37,38)(17,22,23,18,20,24,19,21,25) (26,31,33,27,29,34,28,30,32) Orbits under the action of the full automorphism group (2,10,12)(3,9,13)(4,8,11)(5,6,7)(14,15,16,35,36,37,38,39,40)(17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34) Distribution of neighbour graphs: ( 9, 18, 0, 9, 4) The 2-(40,13,4) design has no further polarities. -------------------------------------------------------------------------- Number 20. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000010001001100001100100 0100010010010001000001100100010100010010 0100001001001100100000001001010001010010 0100001001001010010000100100001100001001 0100001001001001001000010010100010100100 0010100001010100001010000010100001010001 0010100001010010100001000001001100100010 0010100001010001010100000100010010001100 0010010100001100001010001000001010100010 0010010100001010100001100000010001001100 0010010100001001010100010000100100010001 0010001010100100010001100001000010001010 0010001010100010001100001010000100100001 0010001010100001100010010100000001010100 0001100010001100001010010001010000001100 0001100010001010100001001100100000010001 0001100010001001010100100010001000100010 0001010001100100010001010100010001000001 0001010001100010001100100001001010000100 0001010001100001100010001010100100000010 0001001100010100010001001010001100010000 0001001100010010001100010100100001001000 0001001100010001100010100001010010100000 Order of automorphism group: 4 Generators of the automorphism group 1. (5,13)(6,12)(7,11)(8,10)(14,21)(15,20)(16,22)(17,18)(23,26)(24,28) (25,27)(29,31)(33,34)(35,40)(36,39)(37,38) 2. (3,4)(5,7)(8,10)(11,13)(14,20)(15,21)(16,22)(23,39)(24,40)(25,38) (26,36)(27,37)(28,35)(29,34)(30,32)(31,33) Orbits under the action of the full automorphism group (3,4)(5,7,11,13)(6,12)(8,10)(14,15,20,21)(16,22)(17,18)(23,26,36,39)(24,28,35,40)(25,27,37,38)(29,31,33,34)(30,32) Distribution of neighbour graphs: ( 4, 20, 4, 10, 2) The 2-(40,13,4) design has polarities with 16 and 28 abs. pts. -------------------------------------------------------------------------- Number 21. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010001010001001 0100010010010010000010001001100001100100 0100010010010001000001100100010100010010 0100001001001100100000001001010001010010 0100001001001010010000100100001100001001 0100001001001001001000010010100010100100 0010100001010100001010000010100001010001 0010100001010010100001000001010100001100 0010100001010001010100000100001010100010 0010010100001100001010001000010010001100 0010010100001010100001100000001001100010 0010010100001001010100010000100100010001 0010001010100100010001100001000010001010 0010001010100010001100010100000001100001 0010001010100001100010001010000100010100 0001100010001100001010010001001000100010 0001100010001010100001001100100000010001 0001100010001001010100100010010000001100 0001010001100100010001001010010100000001 0001010001100010001100100001001010000100 0001010001100001100010010100100001000010 0001001100010100010001010100001001010000 0001001100010010001100001010100100001000 0001001100010001100010100001010010100000 Order of automorphism group: 6 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (3,4)(5,11)(6,12)(7,13)(15,16)(18,19)(21,22)(23,35)(24,37)(25,36) (26,38)(27,40)(28,39)(29,32)(30,34)(31,33) Orbits under the action of the full automorphism group (3,4)(5,11)(6,12)(7,13)(14,15,16)(17,18,19)(20,21,22)(23,24,25,35,36,37)(26,27,28,38,39,40)(29,30,31,32,33,34) Distribution of neighbour graphs: ( 6, 22, 6, 3, 3) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 22. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000001100001001100100010 0100010010010001000010001100100010010100 0100001001001100100000001001001010010010 0100001001001010001000100010100100001001 0100001001001001010000010100010001100100 0010100001010100010010000100001001010001 0010100001010010100001000001100010001100 0010100001010001001100000010010100100010 0010010100001100001001100000010010001010 0010010100001010100010001000001001100100 0010010100001001010100010000100100010001 0010001010100100001010010001000001100010 0010001010100010100001001100000100010001 0010001010100001010100100010000010001100 0001100010001100010010001001010000001100 0001100010001010001100010100001000100001 0001100010001001100001100010100000010010 0001010001100100010001001010100010000001 0001010001100010001100100001010001000100 0001010001100001100010010100001100000010 0001001100010100001001010010001100010000 0001001100010010010100001100100001001000 0001001100010001100010100001010010100000 Order of automorphism group: 4 Generators of the automorphism group 1. (5,8)(6,10)(7,9)(12,13)(15,16)(17,20)(18,22)(19,21)(23,26)(24,28) (25,27)(30,31)(32,38)(33,40)(34,39)(36,37) 2. (3,4)(6,7)(9,10)(12,13)(17,20)(18,21)(19,22)(23,32)(24,33)(25,34) (26,38)(27,39)(28,40)(29,35)(30,36)(31,37) Orbits under the action of the full automorphism group (3,4)(5,8)(6,7,9,10)(12,13)(15,16)(17,20)(18,19,21,22)(23,26,32,38)(24,28,33,40)(25,27,34,39)(29,35)(30,31,36,37) Distribution of neighbour graphs: (12, 12, 0, 14, 2) The 2-(40,13,4) design has polarities with 16 and 28 abs. pts. -------------------------------------------------------------------------- Number 23. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000001100001001100100010 0100010010010001000010001100100010010100 0100001001001100100000001001001010010010 0100001001001010001000100100010001100001 0100001001001001010000010010100100001100 0010100001010100010010000001100010001001 0010100001010010100001000100001001010100 0010100001010001001100000010010100100010 0010010100001100001010010000001100001010 0010010100001010100001001000100010100001 0010010100001001010100100000010001010100 0010001010100100001001100010000001010010 0010001010100010100010001001000100001100 0010001010100001010100010100000010100001 0001100010001100010001001100010000010001 0001100010001010001100100010001000001100 0001100010001001100010010001100000100010 0001010001100100010010001010001001000100 0001010001100010001100010001100100000001 0001010001100001100001100100010010000010 0001001100010100001001010001010010100000 0001001100010010010100001100100001001000 0001001100010001100010100010001100010000 Order of automorphism group: 6 Generators of the automorphism group 1. (3,4)(5,8)(6,9)(7,10)(15,16)(18,19)(21,22)(23,38)(24,40)(25,39)(26,32) (27,34)(28,33)(29,35)(30,37)(31,36) 2. (2,3)(5,13)(6,11)(7,12)(14,26)(15,28)(16,27)(17,31)(18,30)(19,29) (20,24)(21,23)(22,25)(33,34)(35,37)(38,39) Orbits under the action of the full automorphism group (2,3,4)(5,8,13)(6,9,11)(7,10,12)(14,26,32)(15,16,27,28,33,34)(17,31,36)(18,19,29,30,35,37)(20,24,40)(21,22,23,25,38,39) Distribution of neighbour graphs: (18, 18, 0, 3, 1) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 24. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000001100001001100100010 0100010010010001000010001100100010010100 0100001001001100100000001001001010010010 0100001001001010001000100100010100001001 0100001001001001010000010010100001100100 0010100001010100010010000001100001010001 0010100001010010100001000100001010001100 0010100001010001001100000010010100100010 0010010100001100001010010000001001100010 0010010100001010100001001000100100010001 0010010100001001010100100000010010001100 0010001010100100001001100010000010001010 0010001010100010100010001001000001100100 0010001010100001010100010100000100010001 0001100010001100010010001010001000001100 0001100010001010001100010001100000100001 0001100010001001100001100100010000010010 0001010001100100010001001100010010000001 0001010001100010001100100010001001000100 0001010001100001100010010001100100000010 0001001100010100001001010001010100010000 0001001100010010010100001100100001001000 0001001100010001100010100010001010100000 Order of automorphism group: 2 Generators of the automorphism group 1. (2,3)(5,13)(6,11)(7,12)(14,26)(15,28)(16,27)(17,31)(18,30)(19,29) (20,24)(21,23)(22,25)(32,34)(36,37)(38,39) Orbits under the action of the full automorphism group (2,3)(5,13)(6,11)(7,12)(14,26)(15,28)(16,27)(17,31)(18,30)(19,29)(20,24)(21,23)(22,25)(32,34)(36,37)(38,39) Distribution of neighbour graphs: ( 8, 18, 2, 9, 3) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 25. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000010001001001100100100 0100010010010001000001100100100010010010 0100001001001100100000001001001010010010 0100001001001010010000100100100001001001 0100001001001001001000010010010100100100 0010100001010100001010000001010010100001 0010100001010010100001000100001001010100 0010100001010001010100000010100100001010 0010010100001100001010010000001100001010 0010010100001010100001001000100010100001 0010010100001001010100100000010001010100 0010001010100100001010001010000001010100 0010001010100010100001100001000100001010 0010001010100001010100010100000010100001 0001100010001100010001001100010000010001 0001100010001010001100100010001000001100 0001100010001001100010010001100000100010 0001010001100100010001100010001001000010 0001010001100010001100010001100100000001 0001010001100001100010001100010010000100 0001001100010100010001010001100010001000 0001001100010010001100001100010001100000 0001001100010001100010100010001100010000 Order of automorphism group: 648 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (5,6,7)(8,9,10)(11,12,13)(14,17,20)(15,18,21)(16,19,22)(23,27,31) (24,28,29)(25,26,30)(32,37,39)(33,35,40)(34,36,38) 3. (5,8,11)(6,9,12)(7,10,13)(23,26,29)(24,27,30)(25,28,31)(32,38,35) (33,39,36)(34,40,37) 4. (3,4)(8,11)(9,12)(10,13)(15,16)(18,19)(21,22)(23,32)(24,34)(25,33) (26,35)(27,37)(28,36)(29,38)(30,40)(31,39) 5. (2,3)(8,12)(9,13)(10,11)(14,23)(15,25)(16,24)(17,28)(18,27)(19,26) (20,30)(21,29)(22,31)(32,33)(36,37)(38,40) 6. (1,2)(5,14)(6,20)(7,17)(8,16)(9,22)(10,19)(11,15)(12,21)(13,18)(24,29) (25,26)(27,31)(33,35)(34,38)(37,39) Orbits under the action of the full automorphism group (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40) Distribution of neighbour graphs: ( 0, 36, 0, 0, 4) The 2-(40,13,4) design has polarities with 16 abs. pts. -------------------------------------------------------------------------- Number 26. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000010001001001100100100 0100010010010001000001100100100010010010 0100001001001100100000001001001010010010 0100001001001010010000100100100001001001 0100001001001001001000010010010100100100 0010100001010100001010000001010100001010 0010100001010010100001000100001010100001 0010100001010001010100000010100001010100 0010010100001100001010010000001001010100 0010010100001010100001001000100100001010 0010010100001001010100100000010010100001 0010001010100100001010001010000010100001 0010001010100010100001100001000001010100 0010001010100001010100010100000100001010 0001100010001100010001100010001000010001 0001100010001010001100010001100000001100 0001100010001001100010001100010000100010 0001010001100100010001010001100001000010 0001010001100010001100001100010100000001 0001010001100001100010100010001010000100 0001001100010100010001001100010010001000 0001001100010010001100100010001001100000 0001001100010001100010010001100100010000 Order of automorphism group: 51840 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (6,7)(8,11)(9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)(26,29) (27,31)(28,30)(33,34)(35,38)(36,40)(37,39) 3. (5,6)(8,12)(9,11)(10,13)(14,17)(15,19)(16,18)(21,22)(23,27)(24,26) (25,28)(29,30)(32,37)(33,36)(34,35)(38,40) 4. (5,8,11)(6,9,12)(7,10,13)(23,26,29)(24,27,30)(25,28,31)(32,38,35) (33,39,36)(34,40,37) 5. (3,4)(8,11)(9,12)(10,13)(15,16)(18,19)(21,22)(23,32)(24,34)(25,33) (26,35)(27,37)(28,36)(29,38)(30,40)(31,39) 6. (2,3)(8,12)(9,13)(10,11)(14,23)(15,25)(16,24)(17,28)(18,27)(19,26) (20,30)(21,29)(22,31)(33,34)(35,37)(38,39) 7. (2,5)(3,6)(4,7)(9,10)(15,16)(17,23)(18,25)(19,24)(20,32)(21,34)(22,33) (27,28)(29,35)(30,37)(31,36)(39,40) 8. (1,2)(5,14)(6,17)(7,20)(8,15)(9,18)(10,21)(11,16)(12,19)(13,22)(24,26) (25,29)(28,30)(33,38)(34,35)(36,40) Graph is transitive Distribution of neighbour graphs: ( 0, 0, 0, 0, 40) The 2-(40,13,4) design has polarities with 10 and 16 abs. pts. -------------------------------------------------------------------------- Number 27. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000010001001001100100100 0100010010010001000001100100100010010010 0100001001001100100000001001001010010010 0100001001001010010000100100100001001001 0100001001001001001000010010010100100100 0010100001010100001010000010001100010001 0010100001010010100001000001100001100010 0010100001010001010100000100010010001100 0010010100001100001010001000010001100010 0010010100001010100001100000001010001100 0010010100001001010100010000100100010001 0010001010100100001010010001000010001100 0010001010100010100001001100000100010001 0010001010100001010100100010000001100010 0001100010001100010001100001010000001010 0001100010001010001100001010100000100001 0001100010001001100010010100001000010100 0001010001100100010001010100001010000001 0001010001100010001100100001010001000100 0001010001100001100010001010100100000010 0001001100010100010001001010100001010000 0001001100010010001100010100001100001000 0001001100010001100010100001010010100000 Order of automorphism group: 432 Generators of the automorphism group 1. (6,7)(8,11)(9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)(26,29) (27,31)(28,30)(33,34)(35,38)(36,40)(37,39) 2. (5,6)(8,12)(9,11)(10,13)(14,17)(15,19)(16,18)(21,22)(23,27)(24,26) (25,28)(29,30)(32,37)(33,36)(34,35)(38,40) 3. (5,8,11)(6,9,12)(7,10,13)(14,16,15)(17,19,18)(20,22,21)(23,28,30) (24,26,31)(25,27,29)(32,40,36)(33,38,37)(34,39,35) 4. (3,4)(8,11)(9,12)(10,13)(15,16)(18,19)(21,22)(23,32)(24,34)(25,33) (26,35)(27,37)(28,36)(29,38)(30,40)(31,39) 5. (2,3)(8,12)(9,13)(10,11)(14,23)(15,24)(16,25)(17,27)(18,28)(19,26) (20,31)(21,29)(22,30)(33,34)(35,36)(38,40) 6. (1,2)(5,14)(6,17)(7,20)(8,15)(9,18)(10,21)(11,16)(12,19)(13,22)(24,26) (25,29)(28,30)(33,38)(34,35)(36,40) Orbits under the action of the full automorphism group (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40) Distribution of neighbour graphs: ( 0, 0, 0, 36, 4) The 2-(40,13,4) design has polarities with 16 and 28 abs. pts. -------------------------------------------------------------------------- Number 28. 0111111111111000000000000000000000000000 1011000000000111111111000000000000000000 1101000000000000000000111111111000000000 1110000000000000000000000000000111111111 1000011000000111000000111000000111000000 1000101000000000111000000111000000111000 1000110000000000000111000000111000000111 1000000011000111000000000111000000000111 1000000101000000111000000000111111000000 1000000110000000000111111000000000111000 1000000000011111000000000000111000111000 1000000000101000111000111000000000000111 1000000000110000000111000111000111000000 0100100100100000100100100100100100100100 0100100100100000010010010010010010010010 0100100100100000001001001001001001001001 0100010010010100000100010010010001001001 0100010010010010000010001001001100100100 0100010010010001000001100100100010010010 0100001001001100100000001001001010010010 0100001001001010010000100100100001001001 0100001001001001001000010010010100100100 0010100001010100001010000010001100010001 0010100001010010100001000001100010001100 0010100001010001010100000100010001100010 0010010100001100001010001000010010001100 0010010100001010100001100000001001100010 0010010100001001010100010000100100010001 0010001010100100001010010001000001100010 0010001010100010100001001100000100010001 0010001010100001010100100010000010001100 0001100010001100010001100001010000001010 0001100010001010001100010100001000100001 0001100010001001100010001010100000010100 0001010001100100010001001010100010000001 0001010001100010001100100001010001000100 0001010001100001100010010100001100000010 0001001100010100010001010100001001010000 0001001100010010001100001010100100001000 0001001100010001100010100001010010100000 Order of automorphism group: 72 Generators of the automorphism group 1. (14,15,16)(17,18,19)(20,21,22)(23,24,25)(26,27,28)(29,30,31)(32,33,34) (35,36,37)(38,39,40) 2. (6,7)(8,11)(9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)(26,29) (27,31)(28,30)(33,34)(35,38)(36,40)(37,39) 3. (5,8,11)(6,9,12)(7,10,13)(23,26,29)(24,27,30)(25,28,31)(32,38,35) (33,39,36)(34,40,37) 4. (3,4)(8,11)(9,12)(10,13)(15,16)(18,19)(21,22)(23,32)(24,34)(25,33) (26,35)(27,37)(28,36)(29,38)(30,40)(31,39) 5. (1,2)(5,14)(6,17)(7,20)(8,15)(9,18)(10,21)(11,16)(12,19)(13,22)(24,26) (25,29)(28,30)(33,38)(34,35)(36,40) Orbits under the action of the full automorphism group (1,2)(3,4)(5,8,11,14,15,16)(6,7,9,10,12,13,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40) Distribution of neighbour graphs: ( 0, 12, 0, 18, 10) The 2-(40,13,4) design has polarities with 16 and 28 abs. pts. --------------------------------------------------------------------------