### 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 [64,161]. 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 [64,161]. # 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(64,161));' can be # used to create a list e of the elements of the group [64,161]. # In our list of (64,28,12) DSs in group [64,161], we have [1, # 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 28, 29, 30, # 38, 40, 44, 51, 52, 53, 54, 55, 56, 57, 58, 60, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[9], e[12], e[13], e[14], e[18], e[20], e[28], e[29], e[30], e[38], # e[40], e[44], e[51], e[52], e[53], e[54], e[55], e[56], e[57], e[58], e[60], e[64]] # where e is the list of the elements of the group [64,161] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,161));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 161] ### The number of inequivalent (64,28,12) DSs in group [64,161] is 398 ### The list of all inequivalent (64,28,12) DSs in group [64,161] is DS161:=[ [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 28, 29, 30, 38, 40, 44, 51, 52, 53, 54, 55, 56, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 19, 20, 28, 30, 34, 38, 41, 44, 50, 52, 54, 56, 58, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 19, 20, 28, 31, 34, 35, 38, 39, 44, 50, 52, 55, 56, 57, 58, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 19, 20, 23, 28, 31, 33, 35, 38, 39, 44, 45, 50, 52, 54, 56, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 19, 20, 23, 28, 29, 31, 33, 35, 38, 39, 44, 51, 52, 54, 56, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 19, 20, 23, 27, 29, 31, 34, 35, 38, 39, 44, 50, 52, 55, 56, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 19, 20, 28, 31, 35, 38, 41, 44, 50, 52, 53, 56, 58, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 19, 20, 23, 29, 31, 33, 35, 38, 41, 44, 49, 52, 54, 56, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 19, 20, 23, 28, 31, 33, 35, 38, 41, 45, 50, 52, 54, 56, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 19, 20, 23, 27, 29, 31, 35, 38, 41, 44, 50, 52, 53, 56, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 29, 31, 34, 35, 38, 40, 44, 49, 52, 55, 56, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 28, 31, 35, 38, 40, 44, 45, 50, 52, 53, 56, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 28, 31, 34, 35, 38, 40, 45, 50, 52, 55, 56, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 28, 29, 31, 35, 38, 40, 44, 51, 52, 53, 56, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 19, 20, 28, 32, 34, 38, 39, 44, 50, 54, 56, 57, 58, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 19, 20, 23, 28, 32, 34, 38, 39, 43, 45, 50, 54, 56, 57, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 19, 20, 23, 28, 32, 33, 35, 38, 39, 50, 54, 56, 57, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 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27, 28, 32, 34, 37, 40, 42, 43, 46, 49, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 24, 25, 27, 32, 37, 40, 42, 44, 45, 48, 49, 53, 55, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 24, 25, 27, 32, 34, 37, 40, 42, 44, 45, 48, 49, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 24, 25, 27, 28, 32, 37, 40, 42, 45, 48, 51, 53, 55, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 24, 25, 27, 28, 32, 34, 37, 40, 42, 45, 48, 51, 58, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 29, 32, 37, 40, 42, 44, 45, 46, 51, 53, 55, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 29, 32, 34, 37, 40, 42, 44, 45, 46, 51, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 29, 30, 37, 40, 42, 44, 45, 48, 49, 53, 55, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 29, 30, 34, 37, 40, 42, 44, 45, 48, 49, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 28, 29, 32, 37, 40, 42, 45, 46, 49, 53, 55, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 28, 29, 32, 34, 37, 40, 42, 45, 46, 49, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 28, 29, 30, 37, 40, 42, 45, 48, 51, 53, 55, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 23, 24, 28, 29, 30, 34, 37, 40, 42, 45, 48, 51, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 21, 22, 24, 25, 27, 37, 44, 45, 48, 49, 51, 52, 53, 55, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 21, 22, 24, 25, 27, 28, 34, 37, 44, 45, 48, 52, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 21, 22, 23, 24, 25, 34, 37, 44, 45, 48, 50, 51, 52, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 21, 22, 23, 24, 25, 28, 34, 37, 45, 48, 49, 50, 52, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 21, 22, 24, 25, 29, 35, 37, 39, 41, 44, 48, 52, 53, 54, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 18, 21, 22, 24, 25, 29, 35, 37, 48, 52, 53, 54, 57, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 24, 25, 32, 34, 42, 43, 44, 50, 51, 58, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 24, 25, 28, 32, 34, 42, 43, 49, 50, 59, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 24, 25, 27, 32, 42, 44, 45, 49, 51, 53, 55, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 24, 25, 27, 28, 32, 34, 42, 44, 45, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 30, 43, 45, 48, 49, 50, 51, 53, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 28, 30, 43, 44, 45, 48, 50, 53, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 43, 44, 45, 50, 51, 52, 53, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 28, 43, 45, 49, 50, 52, 53, 60, 61, 63, 64 ], [ 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, 13, 21, 22, 24, 25, 29, 34, 35, 37, 40, 41, 48, 52, 53, 57, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 21, 22, 24, 25, 29, 35, 37, 39, 40, 48, 52, 55, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 19, 21, 24, 25, 32, 34, 41, 42, 44, 50, 53, 54, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 19, 21, 22, 24, 30, 39, 41, 48, 50, 53, 54, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 21, 24, 29, 32, 35, 39, 40, 42, 44, 46, 48, 55, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 21, 24, 25, 29, 32, 35, 39, 40, 42, 55, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 19, 21, 24, 25, 32, 39, 42, 44, 50, 54, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 19, 21, 22, 24, 30, 44, 48, 50, 53, 54, 57, 61, 63, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 204 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS161[2], DS161[5], DS161[8], DS161[9], DS161[11], DS161[12], DS161[13], DS161[15], DS161[16], DS161[17], DS161[20], DS161[21], DS161[23], DS161[24], DS161[25], DS161[38], DS161[39], DS161[47], DS161[48], DS161[55], DS161[66], DS161[77], DS161[78], DS161[83], DS161[84], DS161[89], DS161[90], DS161[93], DS161[94], DS161[99], DS161[105], DS161[107], DS161[109], DS161[111], DS161[115], DS161[116], DS161[117], DS161[120], DS161[121], DS161[134], DS161[135], DS161[143], DS161[146], DS161[147], DS161[148], DS161[149], DS161[150], DS161[151], DS161[152], DS161[155], DS161[156], DS161[157], DS161[158], DS161[159], DS161[160], DS161[163], DS161[165], DS161[166], DS161[167], DS161[171], DS161[172], DS161[173], DS161[174], DS161[175], DS161[176], DS161[182], DS161[184], DS161[185], DS161[187], DS161[190], DS161[196], DS161[198], DS161[205], DS161[206], DS161[208], DS161[212], DS161[214], DS161[215], DS161[217], DS161[218], DS161[219], DS161[220], DS161[221], DS161[226], DS161[227], DS161[229], DS161[234], DS161[236], DS161[239], DS161[240], DS161[241], DS161[242], DS161[243], DS161[244], DS161[245], DS161[247], DS161[248], DS161[249], DS161[250], DS161[251], DS161[252], DS161[253], DS161[254], DS161[255], DS161[256], DS161[257], DS161[258], DS161[259], DS161[260], DS161[263], DS161[265], DS161[268], DS161[269], DS161[270], DS161[271], DS161[272], DS161[273], DS161[274], DS161[275], DS161[276], DS161[279], DS161[280], DS161[281], DS161[282], DS161[283], DS161[284], DS161[287], DS161[288], DS161[289], DS161[290], DS161[291], DS161[292], DS161[293], DS161[294], DS161[295], DS161[296], DS161[297], DS161[298], DS161[299], DS161[300], DS161[301], DS161[304], DS161[305], DS161[306], DS161[307], DS161[309], DS161[310], DS161[312], DS161[313], DS161[315], DS161[316], DS161[318], DS161[320], DS161[323], DS161[325], DS161[326], DS161[327], DS161[328], DS161[331], DS161[333], DS161[335], DS161[336], DS161[339], DS161[341], DS161[343], DS161[344], DS161[345], DS161[346], DS161[347], DS161[348], DS161[349], DS161[350], DS161[353], DS161[356], DS161[358], DS161[360], DS161[361], DS161[363], DS161[364], DS161[365], DS161[370], DS161[371], DS161[373], DS161[374], DS161[375], DS161[376], DS161[377], DS161[378], DS161[379], DS161[380], DS161[381], DS161[382], DS161[385], DS161[386], DS161[387], DS161[389], DS161[391], DS161[392], DS161[393], DS161[394], DS161[395], DS161[396], DS161[397], DS161[398]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,161]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,161] is 84 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,161] is RDS161:=[ [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 21, 22, 24, 26, 33, 34, 38, 41, 43, 45, 47, 49, 50, 51, 53, 57, 58, 60 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 21, 22, 23, 24, 26, 33, 34, 38, 41, 44, 47, 49, 50, 51, 53, 57, 59, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 24, 26, 33, 38, 39, 43, 45, 47, 49, 50, 51, 55, 58, 60, 63 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 33, 38, 39, 44, 47, 49, 50, 51, 55, 59, 63, 64 ], [ 2, 4, 5, 6, 9, 11, 13, 15, 17, 18, 21, 22, 23, 31, 33, 36, 38, 39, 44, 45, 49, 50, 51, 52, 55, 56, 60, 63 ], [ 2, 4, 5, 6, 9, 11, 12, 13, 15, 17, 18, 21, 22, 23, 32, 33, 36, 38, 39, 44, 49, 50, 51, 55, 56, 59, 63, 64 ], [ 2, 4, 5, 6, 8, 9, 11, 13, 17, 21, 22, 23, 24, 26, 31, 33, 34, 41, 44, 45, 47, 49, 50, 51, 52, 53, 57, 60 ], [ 2, 4, 5, 6, 8, 9, 11, 13, 17, 18, 21, 22, 23, 24, 26, 31, 33, 39, 44, 45, 47, 49, 50, 51, 52, 55, 60, 63 ], [ 2, 4, 5, 6, 8, 9, 11, 12, 13, 17, 21, 22, 23, 24, 26, 32, 33, 34, 41, 44, 47, 49, 50, 51, 53, 57, 59, 64 ], [ 2, 4, 5, 6, 8, 9, 11, 12, 13, 17, 18, 21, 22, 23, 24, 26, 32, 33, 39, 44, 47, 49, 50, 51, 55, 59, 63, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 15, 16, 17, 23, 24, 26, 29, 31, 33, 34, 38, 41, 44, 47, 49, 52, 53, 57, 59, 62, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 15, 16, 17, 18, 23, 24, 26, 29, 31, 33, 38, 39, 44, 47, 49, 52, 55, 59, 62, 63, 64 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 16, 17, 24, 26, 32, 33, 34, 36, 41, 43, 45, 47, 49, 50, 51, 53, 56, 57, 58, 60 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 16, 17, 23, 24, 26, 32, 33, 34, 36, 41, 44, 47, 49, 50, 51, 53, 56, 57, 59, 64 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 16, 17, 18, 24, 26, 32, 33, 36, 39, 43, 45, 47, 49, 50, 51, 55, 56, 58, 60, 63 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 16, 17, 18, 23, 24, 26, 32, 33, 36, 39, 44, 47, 49, 50, 51, 55, 56, 59, 63, 64 ], [ 2, 3, 4, 6, 9, 11, 13, 15, 16, 17, 18, 20, 22, 23, 29, 31, 33, 38, 39, 42, 44, 49, 52, 55, 59, 62, 63, 64 ], [ 2, 3, 4, 6, 9, 11, 12, 13, 16, 17, 18, 20, 22, 32, 33, 36, 39, 42, 43, 45, 49, 50, 51, 55, 56, 58, 60, 63 ], [ 2, 3, 4, 6, 9, 11, 12, 13, 16, 17, 18, 20, 22, 23, 32, 33, 36, 39, 42, 44, 49, 50, 51, 55, 56, 59, 63, 64 ], [ 2, 3, 4, 5, 8, 9, 10, 16, 17, 19, 20, 21, 24, 25, 32, 36, 40, 42, 43, 45, 46, 49, 50, 51, 52, 56, 58, 60 ], [ 2, 3, 4, 5, 8, 9, 10, 16, 17, 19, 20, 21, 23, 24, 25, 32, 36, 40, 42, 44, 46, 49, 50, 51, 52, 56, 59, 64 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 19, 20, 21, 24, 30, 36, 40, 42, 43, 45, 48, 49, 50, 51, 56, 58, 60, 61 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 19, 20, 21, 23, 24, 30, 36, 40, 42, 44, 48, 49, 50, 51, 56, 59, 61, 64 ], [ 2, 3, 4, 5, 6, 9, 11, 13, 14, 16, 17, 18, 21, 22, 23, 31, 33, 37, 39, 44, 45, 49, 50, 51, 52, 55, 60, 63 ], [ 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 16, 17, 18, 21, 22, 23, 32, 33, 37, 39, 44, 49, 50, 51, 55, 59, 63, 64 ], [ 2, 3, 4, 5, 6, 9, 10, 12, 13, 15, 16, 17, 21, 22, 31, 33, 34, 38, 41, 43, 45, 49, 50, 51, 53, 57, 58, 60 ], [ 2, 3, 4, 5, 6, 9, 10, 12, 13, 15, 16, 17, 21, 22, 23, 31, 33, 34, 38, 41, 44, 49, 50, 51, 53, 57, 59, 64 ], [ 2, 3, 4, 5, 6, 9, 10, 12, 13, 15, 16, 17, 18, 21, 22, 31, 33, 38, 39, 43, 45, 49, 50, 51, 55, 58, 60, 63 ], [ 2, 3, 4, 5, 6, 9, 10, 12, 13, 15, 16, 17, 18, 21, 22, 23, 31, 33, 38, 39, 44, 49, 50, 51, 55, 59, 63, 64 ], [ 2, 3, 4, 5, 6, 8, 11, 13, 15, 16, 17, 21, 22, 23, 24, 26, 31, 33, 34, 38, 41, 44, 47, 52, 53, 57, 59, 64 ], [ 2, 3, 4, 5, 6, 8, 11, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 31, 33, 38, 39, 44, 47, 52, 55, 59, 63, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 16, 17, 21, 22, 24, 26, 31, 33, 34, 38, 41, 47, 49, 50, 51, 52, 53, 57 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 16, 17, 18, 21, 22, 24, 26, 31, 33, 38, 39, 47, 49, 50, 51, 52, 55, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 21, 22, 24, 34, 35, 38, 43, 45, 49, 50, 51, 53, 54, 55, 58, 60, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 21, 22, 24, 25, 35, 38, 43, 45, 46, 48, 49, 50, 51, 54, 58, 60, 61 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 21, 22, 23, 24, 34, 35, 38, 44, 49, 50, 51, 53, 54, 55, 59, 63, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 21, 22, 23, 24, 25, 35, 38, 44, 46, 48, 49, 50, 51, 54, 59, 61, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 19, 21, 22, 24, 25, 32, 38, 40, 43, 45, 46, 49, 50, 51, 52, 58, 60 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 19, 21, 22, 23, 24, 25, 32, 38, 40, 44, 46, 49, 50, 51, 52, 59, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 18, 19, 21, 22, 24, 38, 39, 40, 43, 45, 49, 50, 51, 55, 58, 60, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 18, 19, 21, 22, 23, 24, 38, 39, 40, 44, 49, 50, 51, 55, 59, 63, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 21, 22, 24, 30, 32, 35, 38, 43, 45, 49, 50, 51, 52, 54, 58, 60 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 21, 22, 23, 24, 30, 32, 35, 38, 44, 49, 50, 51, 52, 54, 59, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 19, 21, 22, 24, 30, 38, 40, 43, 45, 48, 49, 50, 51, 58, 60, 61 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 19, 21, 22, 23, 24, 30, 38, 40, 44, 48, 49, 50, 51, 59, 61, 64 ], [ 1, 3, 4, 6, 7, 8, 9, 13, 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