### 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 [64,131]. 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 [64,131]. # 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(64,131));' can be # used to create a list e of the elements of the group [64,131]. # In our list of (64,28,12) DSs in group [64,131], we have [1, # 2, 3, 4, 5, 8, 9, 12, 13, 15, 16, 22, 24, 25, 27, # 29, 30, 34, 38, 39, 40, 41, 42, 46, 50, 52, 54, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[8], e[9], e[12], e[13], e[15], e[16], e[22], e[24], e[25], e[27], e[29], # e[30], e[34], e[38], e[39], e[40], e[41], e[42], e[46], e[50], e[52], e[54], e[63]] # where e is the list of the elements of the group [64,131] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,131));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 131] ### The number of inequivalent (64,28,12) DSs in group [64,131] is 738 ### The list of all inequivalent (64,28,12) DSs in group [64,131] is DS131:=[ [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 16, 22, 24, 25, 27, 29, 30, 34, 38, 39, 40, 41, 42, 46, 50, 52, 54, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 16, 22, 24, 25, 27, 29, 30, 33, 34, 38, 39, 40, 41, 42, 46, 50, 52, 53 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 14, 15, 19, 21, 23, 32, 33, 36, 39, 43, 44, 49, 50, 51, 52, 53, 54, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 13, 14, 15, 19, 21, 23, 32, 34, 35, 36, 39, 43, 44, 49, 50, 51, 52, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 18, 21, 23, 26, 31, 33, 38, 39, 46, 52, 55, 58, 59, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 18, 21, 23, 26, 31, 33, 38, 39, 43, 46, 52, 55, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 16, 17, 18, 21, 23, 24, 30, 33, 36, 39, 48, 53, 54, 56, 58, 59, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 18, 21, 23, 25, 26, 30, 33, 36, 40, 48, 55, 56, 57, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 18, 21, 23, 25, 26, 30, 33, 36, 40, 43, 48, 55, 56, 57, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 18, 21, 23, 25, 26, 30, 33, 36, 39, 48, 55, 56, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 18, 21, 23, 25, 26, 30, 33, 36, 39, 43, 48, 55, 56, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 18, 21, 23, 24, 30, 34, 35, 36, 39, 48, 55, 56, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 15, 16, 17, 18, 21, 23, 26, 30, 33, 38, 39, 46, 55, 58, 59, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 15, 16, 17, 18, 21, 23, 26, 30, 33, 38, 39, 43, 46, 55, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 18, 21, 26, 30, 38, 39, 43, 46, 53, 54, 55, 56, 58, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 30, 38, 39, 44, 45, 46, 53, 54, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 19, 21, 24, 25, 27, 30, 31, 32, 33, 34, 38, 39, 46, 51, 53, 57, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 12, 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5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 30, 33, 37, 39, 40, 43, 49, 52, 53, 55, 56, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 24, 30, 39, 40, 43, 46, 49, 52, 55, 59, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 19, 30, 34, 35, 37, 46, 49, 52, 53, 55, 56, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 19, 30, 34, 35, 36, 37, 43, 49, 52, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 30, 34, 38, 46, 49, 52, 53, 54, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 27, 34, 36, 38, 39, 42, 47, 49, 50, 54, 55, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 24, 27, 33, 34, 38, 39, 42, 49, 50, 55, 56, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 30, 33, 37, 38, 39, 41, 46, 49, 52, 53, 54, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 30, 34, 37, 38, 39, 40, 46, 49, 52, 53, 55, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 20, 27, 35, 37, 38, 39, 42, 47, 49, 50, 53, 55, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 30, 35, 36, 38, 39, 40, 46, 49, 52, 54, 55, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 30, 33, 35, 38, 39, 40, 43, 49, 52, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 30, 35, 36, 37, 39, 41, 46, 49, 52, 54, 55, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 30, 33, 35, 37, 39, 41, 43, 49, 52, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 30, 33, 38, 39, 41, 43, 49, 52, 54, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 30, 34, 38, 39, 40, 43, 49, 52, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 24, 27, 35, 38, 39, 42, 49, 50, 55, 60, 63, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 366 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS131[10], DS131[14], DS131[18], DS131[19], DS131[20], DS131[22], DS131[29], DS131[30], DS131[31], DS131[32], DS131[37], DS131[38], DS131[39], DS131[41], DS131[53], DS131[54], DS131[55], DS131[58], DS131[61], DS131[63], DS131[64], DS131[67], DS131[69], DS131[81], DS131[85], DS131[91], DS131[92], DS131[93], DS131[94], DS131[95], DS131[96], DS131[97], DS131[98], DS131[101], DS131[102], DS131[104], DS131[110], DS131[111], DS131[122], DS131[125], DS131[126], DS131[128], DS131[129], DS131[131], DS131[140], DS131[144], DS131[148], DS131[152], DS131[162], DS131[164], DS131[168], DS131[173], DS131[180], DS131[185], DS131[193], DS131[194], DS131[195], DS131[197], DS131[203], DS131[204], DS131[205], DS131[210], DS131[212], DS131[215], DS131[216], DS131[225], DS131[226], DS131[227], DS131[228], DS131[230], DS131[231], DS131[232], DS131[236], DS131[237], DS131[242], DS131[243], DS131[248], DS131[251], DS131[252], DS131[253], DS131[254], DS131[255], DS131[256], DS131[258], DS131[266], DS131[270], DS131[272], DS131[273], DS131[275], DS131[276], DS131[277], DS131[278], DS131[279], DS131[280], DS131[282], DS131[283], DS131[289], DS131[291], DS131[295], DS131[298], DS131[302], DS131[303], DS131[304], DS131[305], DS131[306], DS131[307], DS131[308], DS131[309], DS131[310], DS131[314], DS131[316], DS131[317], DS131[319], DS131[323], DS131[324], DS131[325], DS131[326], DS131[327], DS131[331], DS131[332], DS131[333], DS131[338], DS131[341], DS131[342], DS131[343], DS131[345], DS131[346], DS131[348], DS131[349], DS131[350], DS131[351], DS131[352], DS131[354], DS131[357], DS131[362], DS131[363], DS131[364], DS131[365], DS131[366], DS131[373], DS131[374], DS131[376], DS131[377], DS131[378], DS131[381], DS131[387], DS131[396], DS131[397], DS131[399], DS131[403], DS131[405], DS131[408], DS131[411], DS131[412], DS131[415], DS131[416], DS131[417], DS131[418], DS131[419], DS131[422], DS131[423], DS131[428], DS131[430], DS131[431], DS131[433], DS131[434], DS131[435], DS131[440], DS131[445], DS131[446], DS131[447], DS131[448], DS131[449], DS131[450], DS131[453], DS131[455], DS131[459], DS131[462], DS131[465], DS131[467], DS131[473], DS131[474], DS131[475], DS131[476], DS131[479], DS131[480], DS131[481], DS131[482], DS131[483], DS131[484], DS131[485], DS131[486], DS131[487], DS131[488], DS131[489], DS131[492], DS131[494], DS131[495], DS131[496], DS131[499], DS131[501], DS131[503], DS131[510], DS131[513], DS131[514], DS131[517], DS131[520], DS131[521], DS131[523], DS131[526], DS131[536], DS131[537], DS131[538], DS131[539], DS131[544], DS131[545], DS131[549], DS131[553], DS131[554], DS131[559], DS131[564], DS131[566], DS131[571], DS131[572], DS131[575], DS131[576], DS131[579], DS131[580], DS131[581], DS131[582], DS131[583], DS131[585], DS131[586], DS131[587], DS131[588], DS131[589], DS131[595], DS131[596], DS131[597], DS131[598], DS131[601], DS131[603], DS131[604], DS131[605], DS131[606], DS131[608], DS131[609], DS131[610], DS131[611], DS131[612], DS131[614], DS131[615], DS131[616], DS131[617], DS131[618], DS131[620], DS131[621], DS131[622], DS131[624], DS131[626], DS131[627], DS131[628], DS131[629], DS131[630], DS131[631], DS131[632], DS131[633], DS131[634], DS131[635], DS131[636], DS131[637], DS131[638], DS131[639], DS131[640], DS131[641], DS131[642], DS131[643], DS131[644], DS131[645], DS131[646], DS131[647], DS131[648], DS131[649], DS131[650], DS131[651], DS131[654], DS131[656], DS131[658], DS131[659], DS131[660], DS131[661], DS131[662], DS131[663], DS131[664], DS131[665], DS131[666], DS131[667], DS131[668], DS131[669], DS131[670], DS131[671], DS131[672], DS131[673], DS131[674], DS131[675], DS131[676], DS131[677], DS131[678], DS131[679], DS131[680], DS131[681], DS131[682], DS131[683], DS131[684], DS131[685], DS131[686], DS131[687], DS131[688], DS131[689], DS131[690], DS131[691], DS131[692], DS131[693], DS131[694], DS131[696], DS131[697], DS131[698], DS131[699], DS131[700], DS131[701], DS131[702], DS131[703], DS131[704], DS131[705], DS131[707], DS131[708], DS131[709], DS131[710], DS131[711], DS131[712], DS131[713], DS131[714], DS131[715], DS131[716], DS131[717], DS131[718], DS131[719], DS131[720], DS131[721], DS131[722], DS131[723], DS131[724], DS131[725], DS131[726], DS131[727], DS131[728], DS131[729], DS131[730], DS131[731], DS131[732], DS131[733], DS131[734], DS131[735], DS131[736], DS131[737], DS131[738]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,131]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,131] is 216 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,131] is RDS131:=[ [ 3, 5, 6, 8, 9, 13, 14, 15, 17, 21, 22, 23, 24, 26, 39, 40, 44, 47, 49, 50, 51, 54, 55, 56, 57, 59, 63, 64 ], [ 3, 5, 6, 8, 9, 13, 14, 15, 17, 21, 22, 23, 24, 26, 33, 39, 40, 44, 47, 49, 50, 51, 53, 55, 56, 57, 59, 64 ], [ 3, 4, 5, 8, 9, 13, 14, 15, 17, 19, 20, 21, 25, 40, 42, 43, 44, 47, 49, 50, 51, 54, 55, 56, 59, 60, 61, 63 ], [ 3, 4, 5, 8, 9, 13, 14, 15, 17, 19, 20, 21, 25, 33, 40, 42, 43, 44, 47, 49, 50, 51, 53, 55, 56, 59, 60, 61 ], [ 3, 4, 5, 8, 9, 13, 14, 15, 17, 19, 20, 21, 23, 25, 40, 42, 45, 47, 49, 50, 51, 54, 55, 56, 58, 61, 63, 64 ], [ 3, 4, 5, 8, 9, 13, 14, 15, 17, 19, 20, 21, 23, 25, 33, 40, 42, 45, 47, 49, 50, 51, 53, 55, 56, 58, 61, 64 ], [ 3, 4, 5, 7, 8, 9, 13, 14, 15, 17, 18, 21, 22, 28, 29, 34, 37, 40, 44, 45, 46, 47, 48, 51, 53, 58, 59, 63 ], [ 3, 4, 5, 7, 8, 9, 13, 14, 15, 17, 18, 21, 22, 23, 28, 29, 34, 37, 40, 43, 46, 47, 48, 51, 53, 60, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 17, 19, 21, 22, 23, 28, 29, 34, 36, 37, 38, 40, 44, 46, 47, 48, 51, 54, 59, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 17, 19, 21, 22, 23, 28, 29, 33, 34, 36, 37, 38, 40, 44, 46, 47, 48, 51, 53, 59, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 17, 19, 21, 22, 28, 29, 33, 34, 37, 40, 44, 45, 46, 47, 48, 51, 53, 56, 58, 59 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 17, 19, 21, 22, 23, 28, 29, 33, 34, 37, 40, 43, 46, 47, 48, 51, 53, 56, 60, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 17, 19, 21, 22, 23, 25, 28, 29, 37, 40, 43, 47, 51, 53, 54, 55, 56, 60, 61, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 17, 19, 21, 22, 23, 25, 28, 29, 33, 37, 40, 43, 47, 51, 55, 56, 60, 61, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 17, 19, 21, 22, 23, 28, 29, 34, 38, 40, 44, 46, 47, 48, 51, 54, 56, 59, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 17, 19, 21, 22, 23, 28, 29, 33, 34, 38, 40, 44, 46, 47, 48, 51, 53, 56, 59, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 18, 19, 21, 22, 23, 24, 26, 41, 44, 47, 49, 50, 51, 54, 55, 56, 59, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 18, 19, 21, 22, 23, 24, 26, 33, 41, 44, 47, 49, 50, 51, 53, 55, 56, 59, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 17, 19, 21, 22, 28, 29, 34, 36, 40, 44, 45, 46, 47, 48, 51, 54, 58, 59, 63 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 17, 19, 21, 22, 23, 28, 29, 34, 36, 40, 43, 46, 47, 48, 51, 54, 60, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 17, 19, 21, 22, 23, 25, 28, 29, 36, 40, 43, 47, 51, 53, 54, 55, 60, 61, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 14, 15, 17, 19, 21, 22, 23, 25, 28, 29, 33, 36, 40, 43, 47, 51, 55, 60, 61, 63, 64 ], [ 2, 3, 8, 9, 10, 12, 13, 14, 17, 25, 27, 31, 33, 34, 36, 38, 39, 40, 43, 44, 47, 51, 57, 59, 60, 61, 62, 63 ], [ 2, 3, 8, 9, 10, 12, 13, 14, 17, 23, 25, 27, 31, 33, 34, 36, 38, 39, 40, 45, 47, 51, 57, 58, 61, 62, 63, 64 ], [ 2, 3, 8, 9, 10, 12, 13, 14, 15, 17, 23, 25, 31, 39, 40, 45, 47, 49, 50, 51, 54, 55, 56, 57, 58, 61, 63, 64 ], [ 2, 3, 8, 9, 10, 12, 13, 14, 15, 17, 23, 25, 31, 33, 39, 40, 45, 47, 49, 50, 51, 53, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 5, 8, 9, 10, 12, 13, 14, 20, 21, 25, 27, 31, 33, 34, 36, 38, 42, 43, 44, 47, 51, 59, 60, 61, 62, 63 ], [ 2, 3, 5, 8, 9, 10, 12, 13, 14, 20, 21, 23, 25, 27, 31, 33, 34, 36, 38, 42, 45, 47, 51, 58, 61, 62, 63, 64 ], [ 2, 3, 5, 8, 9, 10, 12, 13, 14, 15, 20, 21, 23, 25, 31, 42, 45, 47, 49, 50, 51, 54, 55, 56, 58, 61, 63, 64 ], [ 2, 3, 5, 8, 9, 10, 12, 13, 14, 15, 20, 21, 23, 25, 31, 33, 42, 45, 47, 49, 50, 51, 53, 55, 56, 58, 61, 64 ], [ 2, 3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 21, 22, 23, 31, 33, 39, 40, 44, 49, 50, 51, 53, 55, 56, 57, 59, 64 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 17, 21, 22, 31, 33, 34, 38, 39, 40, 43, 45, 46, 47, 48, 53, 56, 57, 58, 60 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 17, 21, 22, 23, 31, 34, 38, 39, 40, 44, 46, 47, 48, 54, 56, 57, 59, 63, 64 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 17, 21, 22, 23, 31, 33, 34, 38, 39, 40, 44, 46, 47, 48, 53, 56, 57, 59, 64 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 15, 17, 21, 22, 25, 31, 36, 39, 40, 44, 45, 47, 53, 54, 55, 57, 58, 59, 61 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 15, 17, 21, 22, 23, 31, 34, 36, 39, 40, 43, 46, 47, 48, 54, 57, 60, 63, 64 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 15, 17, 21, 22, 23, 25, 31, 36, 39, 40, 43, 47, 53, 54, 55, 57, 60, 61, 64 ], [ 2, 3, 5, 6, 8, 10, 12, 13, 14, 15, 17, 21, 22, 23, 25, 31, 33, 36, 39, 40, 43, 47, 55, 57, 60, 61, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 11, 12, 13, 14, 15, 21, 22, 23, 24, 26, 32, 44, 47, 49, 50, 51, 54, 55, 56, 59, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 11, 12, 13, 14, 15, 21, 22, 23, 24, 26, 32, 33, 44, 47, 49, 50, 51, 53, 55, 56, 59, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 21, 22, 28, 29, 31, 34, 36, 37, 38, 43, 45, 46, 47, 48, 51, 54, 58, 60, 63 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 21, 22, 23, 28, 29, 31, 34, 36, 37, 38, 44, 46, 47, 48, 51, 54, 59, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 21, 22, 23, 28, 29, 31, 33, 34, 36, 37, 38, 44, 46, 47, 48, 51, 53, 59, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 21, 22, 25, 28, 29, 31, 33, 37, 44, 45, 47, 51, 55, 56, 58, 59, 61, 63 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 21, 22, 23, 28, 29, 31, 33, 34, 37, 43, 46, 47, 48, 51, 53, 56, 60, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 21, 22, 23, 25, 28, 29, 31, 37, 43, 47, 51, 53, 54, 55, 56, 60, 61, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 21, 22, 23, 25, 28, 29, 31, 33, 37, 43, 47, 51, 55, 56, 60, 61, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 21, 22, 28, 29, 31, 33, 34, 38, 43, 45, 46, 47, 48, 51, 53, 56, 58, 60 ], [ 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 21, 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