### 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 [64,184]. 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 [64,184]. # 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(64,184));' can be # used to create a list e of the elements of the group [64,184]. # In our list of (64,28,12) DSs in group [64,184], we have [1, # 2, 3, 4, 5, 6, 8, 12, 13, 15, 17, 23, 26, 28, 36, # 37, 39, 41, 42, 43, 45, 51, 54, 55, 58, 59, 63, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[12], e[13], e[15], e[17], e[23], e[26], e[28], e[36], e[37], # e[39], e[41], e[42], e[43], e[45], e[51], e[54], e[55], e[58], e[59], e[63], e[64]] # where e is the list of the elements of the group [64,184] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,184));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 184] ### The number of inequivalent (64,28,12) DSs in group [64,184] is 444 ### The list of all inequivalent (64,28,12) DSs in group [64,184] is DS184:=[ [ 1, 2, 3, 4, 5, 6, 8, 12, 13, 15, 17, 23, 26, 28, 36, 37, 39, 41, 42, 43, 45, 51, 54, 55, 58, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 12, 13, 15, 17, 23, 26, 27, 28, 36, 37, 39, 41, 42, 45, 49, 50, 51, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 12, 13, 15, 17, 23, 24, 26, 28, 36, 37, 39, 41, 42, 45, 46, 47, 51, 54, 55, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 17, 24, 26, 27, 28, 32, 34, 37, 38, 39, 41, 42, 47, 50, 51, 53, 54, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 15, 20, 26, 28, 30, 32, 33, 37, 39, 40, 41, 51, 52, 55, 56, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 15, 17, 24, 26, 27, 28, 32, 36, 37, 39, 41, 42, 47, 50, 51, 54, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 15, 17, 23, 24, 26, 32, 36, 37, 39, 41, 42, 45, 47, 54, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 20, 26, 30, 32, 34, 38, 39, 40, 41, 44, 49, 52, 54, 56, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 20, 26, 28, 30, 32, 34, 38, 39, 40, 41, 51, 52, 54, 56, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 20, 24, 26, 32, 34, 38, 39, 40, 41, 43, 44, 47, 54, 56, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 20, 21, 25, 27, 28, 29, 31, 32, 33, 36, 38, 42, 45, 50, 54, 55, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 20, 22, 29, 34, 36, 37, 42, 43, 45, 46, 47, 50, 54, 55, 56, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 20, 22, 25, 27, 28, 29, 31, 32, 34, 36, 37, 42, 45, 50, 54, 55, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 18, 22, 24, 25, 28, 29, 30, 31, 32, 37, 41, 45, 47, 52, 54, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 17, 20, 22, 29, 37, 38, 42, 43, 45, 46, 47, 50, 53, 54, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 17, 20, 22, 25, 27, 28, 29, 31, 32, 37, 38, 42, 45, 50, 53, 54, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 14, 21, 22, 25, 27, 28, 29, 31, 32, 34, 36, 37, 42, 45, 50, 53, 55, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 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22, 26, 29, 44, 46, 47, 50, 52, 54, 59, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 26, 29, 30, 44, 47, 49, 50, 54, 58, 59, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 26, 29, 30, 44, 46, 47, 49, 50, 54, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 24, 26, 29, 43, 44, 50, 52, 54, 58, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 24, 26, 29, 43, 44, 46, 50, 52, 54, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 24, 26, 29, 30, 43, 44, 49, 50, 54, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 22, 24, 26, 29, 30, 43, 44, 46, 49, 50, 54, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 22, 33, 39, 40, 42, 44, 48, 51, 52, 56, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 22, 25, 28, 33, 40, 42, 44, 52, 53, 56, 57, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 22, 25, 28, 33, 36, 40, 42, 44, 52, 57, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 20, 22, 25, 28, 33, 39, 40, 44, 52, 53, 56, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 20, 22, 25, 28, 33, 36, 39, 40, 44, 52, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 19, 20, 22, 25, 28, 30, 33, 36, 40, 44, 53, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 19, 22, 25, 28, 30, 42, 44, 53, 54, 56, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 19, 20, 22, 36, 39, 44, 48, 51, 52, 54, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 25, 28, 31, 38, 41, 42, 45, 46, 49, 53, 54, 58, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 25, 28, 31, 36, 38, 39, 41, 42, 45, 53, 54, 56, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 25, 28, 31, 38, 41, 42, 43, 45, 47, 49, 53, 54, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 25, 27, 28, 31, 38, 41, 42, 45, 47, 50, 53, 54, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 22, 25, 28, 31, 36, 38, 40, 41, 42, 43, 45, 53, 54, 56, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 21, 22, 28, 31, 36, 37, 38, 41, 43, 44, 45, 48, 53, 57, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 21, 22, 28, 31, 33, 36, 37, 38, 41, 44, 45, 48, 54, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 28, 43, 47, 48, 49, 50, 52, 53, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 28, 30, 43, 46, 47, 48, 50, 53, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 27, 44, 47, 48, 49, 51, 52, 53, 58, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 27, 30, 44, 46, 47, 48, 51, 53, 59, 62, 63, 64 ], [ 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, 16, 19, 21, 22, 23, 25, 27, 46, 47, 49, 51, 52, 53, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 25, 27, 30, 47, 51, 53, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 44, 46, 48, 50, 51, 52, 53, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 30, 44, 48, 49, 50, 51, 53, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 28, 46, 48, 49, 50, 52, 53, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 28, 30, 48, 50, 53, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 22, 25, 28, 31, 37, 38, 39, 41, 42, 43, 45, 53, 54, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 18, 21, 25, 28, 31, 34, 36, 40, 41, 42, 43, 45, 53, 56, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 18, 21, 25, 28, 31, 33, 34, 40, 41, 42, 45, 46, 53, 54, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 22, 25, 28, 31, 37, 38, 40, 41, 42, 45, 53, 54, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 22, 23, 28, 31, 36, 39, 40, 42, 48, 54, 58, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 20, 22, 23, 28, 31, 40, 48, 53, 54, 56, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 21, 25, 28, 31, 34, 38, 39, 40, 42, 43, 45, 53, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 21, 25, 28, 31, 33, 34, 38, 40, 42, 45, 49, 53, 54, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 28, 31, 36, 40, 43, 44, 45, 48, 53, 57, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 28, 31, 33, 36, 40, 44, 45, 48, 54, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 21, 25, 28, 31, 33, 36, 40, 42, 45, 49, 53, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 21, 25, 28, 31, 33, 36, 39, 40, 42, 45, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 21, 25, 28, 31, 34, 35, 39, 40, 42, 45, 46, 53, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 21, 25, 28, 31, 34, 35, 36, 40, 42, 45, 49, 53, 56, 60, 62 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 263 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS184[9], DS184[11], DS184[12], DS184[13], DS184[14], DS184[15], DS184[17], DS184[19], DS184[27], DS184[28], DS184[29], DS184[30], DS184[32], DS184[53], DS184[54], DS184[55], DS184[56], DS184[62], DS184[63], DS184[64], DS184[66], DS184[67], DS184[68], DS184[70], DS184[72], DS184[73], DS184[74], DS184[75], DS184[76], DS184[77], DS184[78], DS184[79], DS184[80], DS184[81], DS184[82], DS184[83], DS184[85], DS184[86], DS184[89], DS184[91], DS184[92], DS184[96], DS184[97], DS184[98], DS184[99], DS184[100], DS184[103], DS184[104], DS184[107], DS184[110], DS184[111], DS184[112], DS184[113], DS184[116], DS184[118], DS184[119], DS184[120], DS184[137], DS184[138], DS184[140], DS184[142], DS184[147], DS184[150], DS184[165], DS184[169], DS184[170], DS184[172], DS184[175], DS184[176], DS184[179], DS184[180], DS184[181], DS184[184], DS184[185], DS184[186], DS184[187], DS184[188], DS184[189], DS184[190], DS184[192], DS184[194], DS184[195], DS184[196], DS184[200], DS184[201], DS184[203], DS184[204], DS184[205], DS184[217], DS184[219], DS184[233], DS184[234], DS184[235], DS184[236], DS184[237], DS184[238], DS184[239], DS184[240], DS184[241], DS184[242], DS184[243], DS184[244], DS184[245], DS184[246], DS184[247], DS184[248], DS184[249], DS184[250], DS184[253], DS184[254], DS184[255], DS184[256], DS184[257], DS184[258], DS184[259], DS184[260], DS184[261], DS184[262], DS184[263], DS184[264], DS184[265], DS184[266], DS184[267], DS184[268], DS184[269], DS184[270], DS184[271], DS184[272], DS184[274], DS184[275], DS184[277], DS184[278], DS184[279], DS184[280], DS184[281], DS184[282], DS184[283], DS184[285], DS184[286], DS184[287], DS184[288], DS184[289], DS184[290], DS184[291], DS184[292], DS184[294], DS184[296], DS184[297], DS184[298], DS184[299], DS184[300], DS184[301], DS184[302], DS184[303], DS184[304], DS184[306], DS184[310], DS184[313], DS184[314], DS184[315], DS184[316], DS184[318], DS184[319], DS184[321], DS184[322], DS184[323], DS184[324], DS184[325], DS184[326], DS184[327], DS184[328], DS184[329], DS184[330], DS184[331], DS184[333], DS184[334], DS184[335], DS184[336], DS184[337], DS184[338], DS184[339], DS184[340], DS184[341], DS184[344], DS184[347], DS184[349], DS184[350], DS184[351], DS184[352], DS184[353], DS184[354], DS184[357], DS184[358], DS184[359], DS184[360], DS184[361], DS184[362], DS184[363], DS184[365], DS184[366], DS184[367], DS184[368], DS184[369], DS184[370], DS184[371], DS184[372], DS184[374], DS184[377], DS184[378], DS184[379], DS184[380], DS184[381], DS184[382], DS184[383], DS184[384], DS184[385], DS184[386], DS184[387], DS184[388], DS184[389], DS184[390], DS184[391], DS184[392], DS184[393], DS184[394], DS184[395], DS184[396], DS184[397], DS184[398], DS184[399], DS184[400], DS184[401], DS184[402], DS184[403], DS184[404], DS184[405], DS184[406], DS184[407], DS184[408], DS184[409], DS184[410], DS184[411], DS184[412], DS184[413], DS184[414], DS184[415], DS184[416], DS184[417], DS184[418], DS184[422], DS184[432], DS184[433], DS184[434], DS184[435], DS184[436], DS184[437], DS184[438], DS184[439], DS184[440], DS184[441], DS184[442], DS184[443], DS184[444]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,184]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,184] is 32 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,184] is RDS184:=[ [ 2, 4, 6, 8, 9, 10, 15, 17, 19, 22, 23, 24, 25, 28, 32, 34, 38, 39, 40, 46, 50, 52, 55, 57, 58, 59, 60, 62 ], [ 2, 4, 6, 8, 9, 10, 14, 15, 19, 22, 23, 28, 32, 34, 36, 37, 38, 43, 44, 46, 47, 48, 50, 52, 55, 56, 58, 62 ], [ 2, 4, 6, 8, 9, 10, 14, 15, 19, 22, 23, 24, 25, 28, 32, 34, 36, 37, 38, 46, 50, 52, 55, 56, 58, 59, 60, 62 ], [ 2, 4, 5, 6, 8, 9, 10, 15, 19, 20, 21, 22, 23, 24, 25, 28, 32, 34, 38, 42, 46, 50, 52, 55, 58, 59, 60, 62 ], [ 2, 3, 6, 8, 9, 10, 16, 18, 22, 23, 28, 32, 33, 34, 41, 43, 44, 46, 47, 48, 50, 52, 53, 54, 55, 58, 62, 63 ], [ 2, 3, 6, 8, 9, 10, 16, 18, 22, 23, 24, 25, 28, 32, 33, 34, 41, 46, 50, 52, 53, 54, 55, 58, 59, 60, 62, 63 ], [ 2, 3, 6, 8, 9, 10, 16, 17, 18, 22, 23, 28, 32, 34, 39, 40, 41, 43, 44, 46, 47, 48, 50, 52, 55, 57, 58, 62 ], [ 2, 3, 6, 8, 9, 10, 16, 17, 18, 22, 23, 24, 25, 28, 32, 34, 39, 40, 41, 46, 50, 52, 55, 57, 58, 59, 60, 62 ], [ 2, 3, 6, 8, 9, 10, 14, 16, 18, 22, 23, 28, 32, 34, 36, 37, 41, 43, 44, 46, 47, 48, 50, 52, 55, 56, 58, 62 ], [ 2, 3, 6, 8, 9, 10, 14, 16, 18, 22, 23, 24, 25, 28, 32, 34, 36, 37, 41, 46, 50, 52, 55, 56, 58, 59, 60, 62 ], [ 2, 3, 5, 6, 8, 9, 10, 16, 18, 20, 21, 22, 23, 28, 32, 34, 41, 42, 43, 44, 46, 47, 48, 50, 52, 55, 58, 62 ], [ 2, 3, 5, 6, 8, 9, 10, 16, 18, 20, 21, 22, 23, 24, 25, 28, 32, 34, 41, 42, 46, 50, 52, 55, 58, 59, 60, 62 ], [ 2, 3, 4, 6, 8, 9, 10, 13, 16, 17, 19, 22, 23, 24, 25, 28, 32, 35, 39, 40, 46, 50, 52, 57, 58, 59, 60, 62 ], [ 2, 3, 4, 6, 8, 9, 10, 13, 14, 16, 19, 22, 23, 28, 32, 35, 36, 37, 43, 44, 46, 47, 48, 50, 52, 56, 58, 62 ], [ 2, 3, 4, 6, 8, 9, 10, 13, 14, 16, 19, 22, 23, 24, 25, 28, 32, 35, 36, 37, 46, 50, 52, 56, 58, 59, 60, 62 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 19, 20, 21, 22, 23, 24, 25, 28, 32, 35, 42, 46, 50, 52, 58, 59, 60, 62 ], [ 1, 2, 7, 8, 9, 10, 15, 17, 18, 23, 24, 25, 28, 32, 34, 38, 39, 40, 41, 46, 50, 52, 55, 57, 58, 59, 60, 62 ], [ 1, 2, 7, 8, 9, 10, 14, 15, 18, 23, 28, 32, 34, 36, 37, 38, 41, 43, 44, 46, 47, 48, 50, 52, 55, 56, 58, 62 ], [ 1, 2, 7, 8, 9, 10, 14, 15, 18, 23, 24, 25, 28, 32, 34, 36, 37, 38, 41, 46, 50, 52, 55, 56, 58, 59, 60, 62 ], [ 1, 2, 5, 7, 8, 9, 10, 15, 18, 20, 21, 23, 24, 25, 28, 32, 34, 38, 41, 42, 46, 50, 52, 55, 58, 59, 60, 62 ], [ 1, 2, 3, 7, 8, 9, 10, 13, 16, 17, 18, 23, 24, 25, 28, 32, 35, 39, 40, 41, 46, 50, 52, 57, 58, 59, 60, 62 ], [ 1, 2, 3, 7, 8, 9, 10, 13, 14, 16, 18, 23, 28, 32, 35, 36, 37, 41, 43, 44, 46, 47, 48, 50, 52, 56, 58, 62 ], [ 1, 2, 3, 7, 8, 9, 10, 13, 14, 16, 18, 23, 24, 25, 28, 32, 35, 36, 37, 41, 46, 50, 52, 56, 58, 59, 60, 62 ], [ 1, 2, 3, 5, 7, 8, 9, 10, 13, 16, 18, 20, 21, 23, 24, 25, 28, 32, 35, 41, 42, 46, 50, 52, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 16, 19, 23, 28, 32, 33, 34, 43, 44, 46, 47, 48, 50, 52, 53, 54, 55, 58, 62, 63 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 16, 19, 23, 24, 25, 28, 32, 33, 34, 46, 50, 52, 53, 54, 55, 58, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 16, 17, 19, 23, 28, 32, 34, 39, 40, 43, 44, 46, 47, 48, 50, 52, 55, 57, 58, 62 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 16, 17, 19, 23, 24, 25, 28, 32, 34, 39, 40, 46, 50, 52, 55, 57, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 14, 16, 19, 23, 28, 32, 34, 36, 37, 43, 44, 46, 47, 48, 50, 52, 55, 56, 58, 62 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 14, 16, 19, 23, 24, 25, 28, 32, 34, 36, 37, 46, 50, 52, 55, 56, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 16, 19, 20, 21, 23, 28, 32, 34, 42, 43, 44, 46, 47, 48, 50, 52, 55, 58, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 16, 19, 20, 21, 23, 24, 25, 28, 32, 34, 42, 46, 50, 52, 55, 58, 59, 60, 62 ] ]; ########################### ########################### ###########################