### This work done by Omar A. AbuGhneim from Jordan University (Email: o_abughneim@yahoo.com). # There are 267 groups of order 64. Among them, there are 259 groups that # admit (64,28,12) difference sets. The other 8 groups do not have any # (64,28,12) difference sets. We wrote programs using the software GAP # to construct all (64,28,12) difference sets in the 259 groups of order # 64 that admit (64,28,12) difference sets. # We refer to the groups as they appear in the SmallGroups library of # the software package GAP. For instance, when we work in groups of # order 64 and write [64,14] we mean group number 14 of order 64 in the # GAP library. # The 8 groups that do not admit (64,28,12) difference sets are # [64, 1], [64, 38], [64, 47], [64, 50], [64, 52], [64, 53], [64, 54], # and [64, 186]. Indeed, these groups are the ones that have either # the cyclic group of order 32 as a factor group or have the dihedral # group of order 32 as a factor group. # Also, we find all (64,28,12) difference sets that are reversible # i.e. all (64,28,12) difference sets D with D=D^(-1). There are 184 # groups of order 64 that have reversible (64,28,12) difference sets. # These groups are [64,i] where i in {2, 3, 4, 5, 8, 9, 10, 11, 12, 13, # 15, 16, 18, 20, 22, 23, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 37, 40, # 41, 42, 43, 45, 46, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, # 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, # 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, # 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 126, # 127, 128, 129, 130, 131, 133, 134, 135, 138, 140, 141, 142, 144, 145, 146, # 147, 149, 150, 151, 152, 153, 155, 156, 157, 159, 160, 161, 162, 163, 164, # 166, 167, 169, 170, 172, 173, 174, 177, 178, 179, 182, 184, 187, 190, 192, # 193, 194, 195, 196, 197, 199, 200, 202, 203, 204, 205, 206, 207, 208, 209, # 210, 211, 213, 214, 215, 216, 219, 221, 222, 226, 227, 231, 232, 234, 236, # 240, 241, 242, 245, 247, 249, 250, 251, 254, 260, 261, 263, 267 }. # We used the software GAP to find all (64,28,12) difference sets and # reversible (64,28,12) difference sets. # In this file we list all inequivalent (64,28,12) difference sets # in group [64,138]. Also, we determine the nonisomorphic # symmetric (64,28,12) designs that rise from these DSs. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,138]. # To determine the nonisomorphic symmetric (64,28,12) designs, we used # the DESIGN package for GAP, Version 1.6, 2011, by L.H. Soicher, # http://designtheory.org/software/gap design/. # Note: The GAP command 'e:=Elements(SmallGroup(64,138));' can be # used to create a list e of the elements of the group [64,138]. # In our list of (64,28,12) DSs in group [64,138], we have [1, # 2, 3, 4, 5, 9, 11, 12, 14, 15, 20, 22, 25, 29, 30, # 32, 33, 35, 40, 41, 48, 52, 55, 56, 57, 58, 63, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[9], e[11], e[12], e[14], e[15], e[20], e[22], e[25], e[29], e[30], e[32], # e[33], e[35], e[40], e[41], e[48], e[52], e[55], e[56], e[57], e[58], e[63], e[64]] # where e is the list of the elements of the group [64,138] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,138));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 138] ### The number of inequivalent (64,28,12) DSs in group [64,138] is 450 ### The list of all inequivalent (64,28,12) DSs in group [64,138] is DS138:=[ [ 1, 2, 3, 4, 5, 9, 11, 12, 14, 15, 20, 22, 25, 29, 30, 32, 33, 35, 40, 41, 48, 52, 55, 56, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 12, 13, 15, 17, 18, 20, 22, 24, 25, 26, 27, 36, 37, 44, 47, 48, 50, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 32, 33, 35, 40, 41, 44, 46, 55, 56, 57, 58, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 30, 32, 33, 35, 40, 41, 44, 49, 52, 55, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 28, 32, 33, 35, 40, 41, 46, 49, 51, 55, 56, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 18, 20, 22, 26, 32, 33, 34, 35, 39, 40, 44, 46, 53, 56, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 18, 20, 22, 26, 28, 32, 33, 34, 35, 39, 40, 46, 49, 51, 53, 56, 61, 62 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 13, 15, 17, 18, 20, 22, 24, 26, 32, 36, 37, 43, 44, 47, 54, 55, 57, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 17, 20, 22, 26, 29, 33, 34, 35, 37, 38, 39, 41, 43, 44, 53, 56, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 15, 17, 18, 20, 22, 26, 29, 33, 34, 35, 36, 37, 43, 44, 53, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 17, 18, 20, 22, 26, 29, 36, 37, 43, 44, 54, 55, 57, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 17, 18, 20, 22, 25, 26, 27, 29, 30, 36, 37, 48, 50, 52, 54, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 16, 18, 19, 21, 31, 32, 36, 41, 43, 44, 45, 47, 48, 50, 51, 52, 56, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 19, 20, 21, 25, 30, 31, 34, 42, 43, 44, 45, 46, 47, 50, 51, 55, 58, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 19, 20, 21, 23, 24, 25, 27, 30, 31, 34, 42, 49, 51, 55, 59, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 18, 19, 21, 24, 25, 30, 31, 36, 41, 44, 45, 50, 51, 56, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 18, 19, 21, 24, 25, 27, 30, 31, 36, 41, 43, 44, 45, 49, 51, 56, 58, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 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46, 52, 53, 54, 56, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 18, 19, 22, 30, 33, 35, 42, 46, 48, 53, 54, 59, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 19, 22, 28, 30, 33, 34, 36, 42, 44, 47, 49, 50, 53, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 18, 19, 30, 32, 34, 35, 37, 46, 52, 53, 55, 56, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 17, 20, 22, 26, 34, 39, 40, 43, 44, 45, 52, 53, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 19, 30, 32, 33, 34, 40, 41, 43, 52, 53, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 30, 32, 33, 35, 41, 43, 44, 49, 52, 53, 54, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 30, 32, 34, 35, 40, 43, 44, 49, 52, 53, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 30, 32, 33, 37, 38, 41, 43, 44, 49, 52, 54, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 17, 30, 32, 34, 37, 38, 40, 43, 44, 49, 52, 55, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 26, 27, 28, 32, 34, 37, 40, 42, 43, 44, 48, 51, 58, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 26, 27, 32, 34, 37, 40, 42, 44, 45, 48, 49, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 22, 24, 27, 31, 34, 35, 36, 38, 39, 40, 42, 45, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 21, 29, 32, 35, 37, 42, 48, 50, 51, 53, 54, 55, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 20, 21, 25, 29, 32, 39, 44, 45, 47, 50, 51, 53, 54, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 20, 21, 23, 25, 29, 32, 33, 39, 47, 50, 51, 58, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 45, 47, 48, 50, 51, 52, 53, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 25, 44, 45, 46, 47, 50, 51, 52, 53, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 25, 30, 44, 45, 47, 49, 50, 51, 53, 58, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 25, 27, 43, 44, 45, 46, 47, 49, 51, 52, 53, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 43, 45, 46, 48, 50, 51, 52, 53, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 28, 43, 44, 45, 46, 48, 49, 50, 52, 53, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 27, 44, 45, 49, 51, 52, 53, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 27, 28, 45, 52, 53, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 25, 43, 46, 47, 50, 51, 52, 53, 58, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 25, 30, 43, 47, 49, 50, 51, 53, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 26, 29, 31, 41, 42, 44, 45, 46, 47, 50, 54, 58, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 26, 27, 29, 31, 33, 41, 42, 44, 45, 46, 47, 49, 53, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 24, 26, 27, 29, 31, 41, 42, 44, 45, 49, 54, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 23, 26, 29, 31, 33, 41, 42, 46, 47, 50, 53, 58, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 21, 23, 29, 32, 33, 36, 38, 40, 41, 42, 43, 45, 57, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 19, 31, 32, 33, 34, 37, 41, 45, 46, 49, 52, 53, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 22, 27, 31, 33, 36, 39, 40, 42, 44, 45, 47, 53, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 27, 32, 34, 40, 42, 43, 45, 46, 47, 49, 53, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 27, 28, 32, 34, 40, 42, 43, 44, 45, 46, 47, 51, 53, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 25, 27, 32, 34, 40, 42, 43, 44, 45, 47, 48, 49, 53, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 25, 27, 28, 32, 34, 40, 42, 43, 45, 47, 48, 51, 53, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 31, 32, 34, 39, 40, 43, 45, 46, 47, 52, 53, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 31, 32, 33, 34, 36, 37, 39, 40, 45, 52, 53, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 25, 31, 32, 34, 39, 40, 43, 44, 45, 47, 48, 52, 53, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 37, 40, 43, 44, 45, 46, 49, 52, 53, 56, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 21, 29, 32, 33, 35, 36, 42, 48, 50, 51, 55, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 21, 27, 32, 34, 35, 38, 40, 42, 45, 46, 49, 53, 55, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 31, 32, 33, 35, 38, 39, 41, 45, 46, 52, 53, 54, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 21, 32, 33, 35, 36, 39, 42, 45, 48, 51, 57, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 19, 31, 32, 33, 34, 38, 40, 45, 46, 49, 52, 53, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 31, 32, 34, 35, 38, 39, 40, 45, 46, 52, 53, 55, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 22, 27, 31, 33, 38, 40, 41, 42, 44, 45, 47, 53, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 21, 27, 32, 33, 34, 35, 36, 38, 40, 42, 45, 49, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 31, 32, 33, 34, 35, 36, 38, 39, 40, 45, 52, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 21, 22, 35, 38, 40, 43, 44, 45, 46, 49, 52, 56, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 31, 32, 33, 34, 35, 36, 37, 39, 41, 45, 52, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 21, 22, 35, 37, 41, 43, 44, 45, 46, 49, 52, 56, 57, 58 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 219 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS138[1], DS138[23], DS138[25], DS138[34], DS138[35], DS138[36], DS138[37], DS138[40], DS138[47], DS138[55], DS138[56], DS138[57], DS138[62], DS138[70], DS138[71], DS138[85], DS138[102], DS138[112], DS138[113], DS138[114], DS138[115], DS138[117], DS138[118], DS138[128], DS138[131], DS138[133], DS138[134], DS138[135], DS138[139], DS138[140], DS138[141], DS138[142], DS138[144], DS138[155], DS138[174], DS138[175], DS138[176], DS138[177], DS138[185], DS138[190], DS138[192], DS138[194], DS138[195], DS138[196], DS138[200], DS138[201], DS138[202], DS138[203], DS138[204], DS138[205], DS138[206], DS138[207], DS138[208], DS138[214], DS138[215], DS138[216], DS138[219], DS138[221], DS138[222], DS138[223], DS138[224], DS138[225], DS138[226], DS138[227], DS138[229], DS138[230], DS138[231], DS138[232], DS138[233], DS138[234], DS138[235], DS138[236], DS138[237], DS138[240], DS138[241], DS138[242], DS138[243], DS138[244], DS138[245], DS138[246], DS138[255], DS138[258], DS138[259], DS138[260], DS138[261], DS138[264], DS138[266], DS138[268], DS138[270], DS138[271], DS138[272], DS138[274], DS138[275], DS138[277], DS138[278], DS138[281], DS138[283], DS138[284], DS138[285], DS138[286], DS138[287], DS138[288], DS138[289], DS138[290], DS138[291], DS138[292], DS138[293], DS138[294], DS138[295], DS138[296], DS138[297], DS138[298], DS138[299], DS138[301], DS138[302], DS138[303], DS138[304], DS138[305], DS138[306], DS138[308], DS138[309], DS138[313], DS138[314], DS138[315], DS138[316], DS138[317], DS138[318], DS138[319], DS138[320], DS138[321], DS138[323], DS138[324], DS138[326], DS138[327], DS138[328], DS138[330], DS138[341], DS138[342], DS138[346], DS138[348], DS138[349], DS138[351], DS138[352], DS138[353], DS138[354], DS138[357], DS138[358], DS138[359], DS138[360], DS138[361], DS138[363], DS138[364], DS138[365], DS138[367], DS138[368], DS138[369], DS138[370], DS138[371], DS138[372], DS138[373], DS138[374], DS138[375], DS138[376], DS138[377], DS138[378], DS138[379], DS138[380], DS138[381], DS138[384], DS138[385], DS138[386], DS138[387], DS138[388], DS138[389], DS138[391], DS138[392], DS138[394], DS138[395], DS138[396], DS138[397], DS138[398], DS138[399], DS138[400], DS138[401], DS138[402], DS138[404], DS138[405], DS138[406], DS138[407], DS138[408], DS138[409], DS138[410], DS138[411], DS138[412], DS138[413], DS138[416], DS138[423], DS138[424], DS138[425], DS138[426], DS138[427], DS138[428], DS138[430], DS138[432], DS138[433], DS138[434], DS138[435], DS138[436], DS138[437], DS138[439], DS138[440], DS138[441], DS138[442], DS138[443], DS138[444], DS138[445], DS138[446], DS138[447], DS138[450]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,138]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,138] is 494 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,138] is RDS138:=[ [ 2, 8, 9, 10, 15, 17, 24, 25, 28, 30, 32, 33, 34, 36, 38, 39, 40, 43, 44, 45, 49, 50, 54, 55, 56, 57, 58, 61 ], [ 2, 8, 9, 10, 11, 15, 17, 24, 25, 27, 28, 33, 34, 36, 38, 39, 40, 45, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62 ], [ 2, 5, 6, 8, 9, 15, 17, 21, 22, 24, 25, 27, 28, 31, 32, 36, 38, 39, 40, 43, 44, 45, 52, 56, 57, 58, 61, 62 ], [ 2, 5, 6, 8, 9, 10, 15, 20, 21, 22, 24, 25, 28, 30, 32, 33, 38, 39, 42, 43, 44, 45, 49, 50, 54, 57, 58, 61 ], [ 2, 5, 6, 8, 9, 10, 15, 17, 21, 22, 24, 25, 28, 30, 32, 36, 38, 39, 40, 43, 44, 45, 49, 50, 56, 57, 58, 61 ], [ 2, 5, 6, 8, 9, 10, 11, 15, 20, 21, 22, 24, 25, 27, 28, 33, 38, 39, 42, 45, 52, 54, 57, 58, 59, 60, 61, 62 ], [ 2, 5, 6, 8, 9, 10, 11, 15, 17, 21, 22, 24, 25, 27, 28, 36, 38, 39, 40, 45, 52, 56, 57, 58, 59, 60, 61, 62 ], [ 2, 3, 8, 9, 11, 16, 17, 24, 25, 28, 30, 31, 34, 36, 39, 40, 45, 49, 50, 53, 55, 56, 57, 58, 59, 60, 61, 63 ], [ 2, 3, 8, 9, 11, 14, 16, 17, 24, 25, 28, 30, 31, 33, 34, 37, 39, 40, 45, 49, 50, 54, 55, 57, 58, 59, 60, 61 ], [ 2, 3, 8, 9, 10, 15, 16, 18, 24, 27, 28, 30, 32, 33, 34, 38, 39, 41, 45, 46, 48, 54, 55, 57, 58, 59, 60, 62 ], [ 2, 3, 8, 9, 10, 15, 16, 18, 24, 25, 28, 30, 32, 33, 34, 38, 39, 41, 43, 44, 45, 49, 50, 54, 55, 57, 58, 61 ], [ 2, 3, 8, 9, 10, 14, 16, 17, 24, 25, 28, 30, 32, 33, 34, 37, 39, 40, 43, 44, 45, 49, 50, 54, 55, 57, 58, 61 ], [ 2, 3, 8, 9, 10, 11, 15, 16, 18, 24, 25, 27, 28, 33, 34, 38, 39, 41, 45, 52, 54, 55, 57, 58, 59, 60, 61, 62 ], [ 2, 3, 8, 9, 10, 11, 14, 16, 17, 24, 25, 27, 28, 33, 34, 37, 39, 40, 45, 52, 54, 55, 57, 58, 59, 60, 61, 62 ], [ 2, 3, 6, 8, 9, 15, 16, 17, 22, 24, 28, 31, 32, 34, 36, 38, 40, 45, 46, 48, 49, 50, 52, 55, 56, 58, 59, 60 ], [ 2, 3, 6, 8, 9, 15, 16, 17, 22, 24, 25, 27, 28, 31, 32, 34, 36, 38, 40, 43, 44, 45, 52, 55, 56, 58, 61, 62 ], [ 2, 3, 6, 8, 9, 14, 15, 16, 22, 24, 28, 31, 32, 34, 37, 38, 39, 45, 46, 48, 49, 50, 52, 55, 57, 58, 59, 60 ], [ 2, 3, 6, 8, 9, 14, 15, 16, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, 39, 43, 44, 45, 52, 55, 57, 58, 61, 62 ], [ 2, 3, 6, 8, 9, 11, 16, 17, 20, 22, 24, 27, 28, 30, 31, 36, 39, 40, 42, 43, 44, 45, 46, 48, 56, 57, 58, 62 ], [ 2, 3, 6, 8, 9, 11, 16, 17, 20, 22, 24, 25, 28, 30, 31, 36, 39, 40, 42, 45, 49, 50, 56, 57, 58, 59, 60, 61 ], [ 2, 3, 6, 8, 9, 11, 15, 16, 20, 22, 24, 27, 28, 30, 31, 33, 34, 38, 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