### 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 [64,152]. 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 [64,152]. # 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(64,152));' can be # used to create a list e of the elements of the group [64,152]. # In our list of (64,28,12) DSs in group [64,152], we have [1, # 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 21, 23, 28, 29, # 35, 36, 38, 40, 41, 44, 51, 52, 53, 57, 58, 59, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[7], e[9], e[10], e[11], e[13], e[14], e[21], e[23], e[28], e[29], e[35], # e[36], e[38], e[40], e[41], e[44], e[51], e[52], e[53], e[57], e[58], e[59], e[63]] # where e is the list of the elements of the group [64,152] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,152));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 152] ### The number of inequivalent (64,28,12) DSs in group [64,152] is 880 ### The list of all inequivalent (64,28,12) DSs in group [64,152] is DS152:=[ [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 21, 23, 28, 29, 35, 36, 38, 40, 41, 44, 51, 52, 53, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 18, 21, 23, 28, 29, 35, 36, 38, 39, 40, 51, 52, 53, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 14, 15, 17, 21, 26, 28, 29, 30, 31, 39, 41, 43, 44, 45, 46, 51, 56, 58, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 14, 15, 17, 18, 21, 26, 28, 29, 30, 31, 43, 45, 46, 51, 56, 57, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 21, 25, 26, 29, 30, 32, 40, 41, 43, 45, 48, 49, 56, 57, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 18, 21, 23, 25, 26, 29, 30, 32, 39, 40, 48, 49, 56, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 17, 21, 25, 26, 29, 30, 32, 39, 41, 45, 48, 49, 56, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 17, 18, 21, 25, 26, 29, 30, 32, 44, 45, 48, 49, 56, 57, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 10, 13, 15, 16, 19, 25, 30, 32, 38, 39, 42, 44, 45, 46, 48, 54, 55, 57, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 10, 13, 15, 16, 19, 25, 30, 32, 33, 38, 39, 42, 43, 44, 45, 46, 48, 53, 55, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 10, 13, 15, 16, 19, 23, 25, 30, 32, 33, 34, 38, 39, 42, 44, 46, 48, 57, 58, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 16, 18, 19, 27, 29, 30, 32, 33, 36, 41, 42, 43, 44, 45, 50, 53, 55, 56, 58 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 15, 16, 19, 27, 29, 30, 32, 33, 38, 39, 42, 43, 44, 45, 50, 53, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 15, 16, 19, 23, 27, 29, 30, 32, 33, 34, 38, 39, 42, 44, 50, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 16, 19, 28, 29, 30, 32, 33, 37, 39, 42, 43, 44, 45, 51, 53, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 16, 19, 23, 28, 29, 30, 32, 33, 34, 37, 39, 42, 44, 51, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 14, 19, 23, 27, 31, 33, 36, 38, 39, 42, 43, 44, 49, 51, 53, 54, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 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53, 55, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 27, 28, 30, 31, 32, 38, 43, 45, 53, 60, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 23, 30, 31, 32, 33, 35, 38, 41, 49, 54, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 19, 30, 31, 32, 33, 34, 38, 40, 45, 53, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 23, 30, 31, 32, 34, 35, 38, 40, 49, 55, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 22, 29, 30, 34, 36, 42, 48, 49, 53, 55, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 22, 29, 30, 34, 36, 42, 43, 48, 49, 53, 55, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 19, 30, 31, 32, 34, 41, 45, 46, 53, 54, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 20, 22, 26, 27, 31, 35, 38, 39, 40, 46, 47, 53, 55, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 24, 26, 27, 30, 32, 35, 37, 48, 51, 52, 53, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 20, 22, 24, 26, 27, 31, 35, 38, 39, 44, 53, 55, 57, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 30, 31, 32, 33, 37, 38, 41, 45, 46, 49, 54, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 21, 22, 25, 33, 41, 45, 46, 47, 49, 52, 53, 54, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 21, 22, 24, 33, 39, 41, 44, 46, 48, 52, 54, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 30, 31, 32, 34, 37, 38, 40, 45, 46, 49, 55, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 21, 22, 25, 34, 40, 45, 46, 47, 49, 52, 53, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 21, 22, 24, 34, 39, 40, 44, 46, 48, 52, 55, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 19, 30, 31, 32, 33, 36, 38, 40, 41, 45, 49, 53, 54, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 19, 22, 24, 26, 31, 33, 41, 42, 44, 50, 53, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 19, 30, 31, 32, 33, 38, 40, 45, 46, 53, 54, 56, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 19, 30, 31, 32, 33, 36, 38, 40, 45, 54, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 30, 31, 32, 34, 35, 36, 38, 40, 45, 49, 54, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 21, 26, 28, 32, 35, 39, 41, 42, 46, 48, 54, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 20, 21, 25, 26, 28, 32, 35, 39, 41, 46, 54, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 19, 30, 31, 32, 34, 36, 37, 40, 41, 45, 49, 53, 55, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 30, 31, 32, 34, 35, 36, 37, 41, 45, 49, 54, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 30, 31, 32, 34, 37, 40, 45, 46, 53, 55, 56, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 30, 31, 32, 34, 36, 37, 40, 45, 55, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 30, 31, 32, 33, 38, 41, 45, 49, 53, 54, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 30, 31, 32, 33, 38, 39, 41, 45, 54, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 30, 31, 32, 34, 38, 40, 45, 49, 53, 55, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 30, 31, 32, 34, 38, 39, 40, 45, 55, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 22, 29, 30, 33, 35, 36, 42, 48, 49, 54, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 22, 29, 30, 33, 35, 36, 42, 43, 48, 49, 54, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 22, 27, 29, 30, 35, 36, 42, 48, 50, 53, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 22, 27, 29, 30, 35, 36, 42, 43, 48, 50, 53, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 30, 34, 37, 43, 46, 52, 53, 54, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 30, 33, 37, 44, 46, 52, 53, 55, 59, 62, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 489 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS152[3], DS152[14], DS152[15], DS152[18], DS152[20], DS152[21], DS152[22], DS152[23], DS152[24], DS152[27], DS152[39], DS152[41], DS152[44], DS152[49], DS152[52], DS152[56], DS152[58], DS152[66], DS152[72], DS152[78], DS152[87], DS152[122], DS152[123], DS152[124], DS152[125], DS152[126], DS152[127], DS152[128], DS152[129], DS152[130], DS152[131], DS152[132], DS152[133], DS152[134], DS152[139], DS152[146], DS152[151], DS152[152], DS152[154], DS152[156], DS152[159], DS152[160], DS152[161], DS152[162], DS152[165], DS152[166], DS152[173], DS152[184], DS152[185], DS152[186], DS152[193], DS152[195], DS152[202], DS152[203], DS152[206], DS152[213], DS152[217], DS152[219], DS152[220], DS152[221], DS152[223], DS152[224], DS152[227], DS152[231], DS152[234], DS152[237], DS152[239], DS152[240], DS152[241], DS152[242], DS152[243], DS152[244], DS152[253], DS152[256], DS152[258], DS152[261], DS152[262], DS152[264], DS152[267], DS152[268], DS152[269], DS152[272], DS152[273], DS152[274], DS152[275], DS152[276], DS152[279], DS152[280], DS152[283], DS152[299], DS152[302], DS152[303], DS152[304], DS152[311], DS152[312], DS152[314], DS152[315], DS152[316], DS152[317], DS152[318], DS152[320], DS152[321], DS152[322], DS152[327], DS152[328], DS152[329], DS152[338], DS152[339], DS152[345], DS152[351], DS152[354], DS152[355], DS152[356], DS152[357], DS152[360], DS152[365], DS152[366], DS152[367], DS152[368], DS152[369], DS152[376], DS152[379], DS152[381], DS152[383], DS152[385], DS152[388], DS152[389], DS152[392], DS152[394], DS152[399], DS152[400], DS152[401], DS152[409], DS152[410], DS152[412], DS152[420], DS152[422], DS152[423], DS152[426], DS152[429], DS152[431], DS152[432], DS152[433], DS152[434], DS152[436], DS152[437], DS152[438], DS152[447], DS152[461], DS152[464], DS152[465], DS152[466], DS152[471], DS152[473], DS152[474], DS152[476], DS152[477], DS152[478], DS152[488], DS152[492], DS152[493], DS152[494], DS152[495], DS152[497], DS152[499], DS152[500], DS152[501], DS152[504], DS152[506], DS152[507], DS152[508], DS152[509], DS152[510], DS152[511], DS152[512], DS152[513], DS152[515], DS152[516], DS152[519], DS152[525], DS152[526], DS152[527], DS152[528], DS152[529], DS152[530], DS152[531], DS152[532], DS152[533], DS152[534], DS152[536], DS152[539], DS152[540], DS152[541], DS152[544], DS152[545], DS152[546], DS152[547], DS152[548], DS152[549], DS152[550], DS152[551], DS152[552], DS152[554], DS152[555], DS152[556], DS152[557], DS152[559], DS152[560], DS152[561], DS152[562], DS152[563], DS152[564], DS152[565], DS152[566], DS152[567], DS152[568], DS152[569], DS152[570], DS152[571], DS152[572], DS152[573], DS152[574], DS152[575], DS152[576], DS152[577], DS152[578], DS152[579], DS152[580], DS152[581], DS152[582], DS152[583], DS152[584], DS152[585], DS152[586], DS152[587], DS152[588], DS152[589], DS152[590], DS152[591], DS152[592], DS152[593], DS152[594], DS152[595], DS152[596], DS152[597], DS152[598], DS152[600], DS152[601], DS152[602], DS152[603], DS152[604], DS152[605], DS152[606], DS152[607], DS152[608], DS152[609], DS152[610], DS152[611], DS152[612], DS152[613], DS152[614], DS152[615], DS152[616], DS152[617], DS152[618], DS152[619], DS152[620], DS152[621], DS152[622], DS152[623], DS152[624], DS152[625], DS152[626], DS152[627], DS152[628], DS152[629], DS152[630], DS152[631], DS152[632], DS152[633], DS152[634], DS152[635], DS152[636], DS152[637], DS152[638], DS152[641], DS152[642], DS152[643], DS152[644], DS152[646], DS152[647], DS152[648], DS152[652], DS152[653], DS152[654], DS152[655], DS152[656], DS152[657], DS152[658], DS152[659], DS152[660], DS152[661], DS152[662], DS152[664], DS152[665], DS152[666], DS152[667], DS152[668], DS152[669], DS152[670], DS152[671], DS152[672], DS152[673], DS152[675], DS152[676], DS152[679], DS152[680], DS152[681], DS152[682], DS152[683], DS152[684], DS152[685], DS152[686], DS152[687], DS152[688], DS152[689], DS152[690], DS152[691], DS152[692], DS152[693], DS152[694], DS152[695], DS152[696], DS152[697], DS152[698], DS152[699], DS152[700], DS152[701], DS152[702], DS152[703], DS152[704], DS152[709], DS152[710], DS152[711], DS152[712], DS152[713], DS152[714], DS152[716], DS152[717], DS152[718], DS152[720], DS152[721], DS152[722], DS152[723], DS152[724], DS152[725], DS152[726], DS152[729], DS152[730], DS152[731], DS152[732], DS152[733], DS152[734], DS152[737], DS152[738], DS152[739], DS152[742], DS152[743], DS152[744], DS152[745], DS152[748], DS152[749], DS152[750], DS152[751], DS152[752], DS152[753], DS152[754], DS152[755], DS152[756], DS152[757], DS152[758], DS152[759], DS152[760], DS152[761], DS152[762], DS152[763], DS152[764], DS152[765], DS152[766], DS152[767], DS152[768], DS152[769], DS152[770], DS152[771], DS152[772], DS152[773], DS152[774], DS152[776], DS152[777], DS152[778], DS152[779], DS152[780], DS152[783], DS152[784], DS152[785], DS152[786], DS152[787], DS152[788], DS152[789], DS152[790], DS152[791], DS152[792], DS152[793], DS152[794], DS152[795], DS152[796], DS152[797], DS152[798], DS152[799], DS152[800], DS152[801], DS152[802], DS152[806], DS152[807], DS152[808], DS152[809], DS152[810], DS152[811], DS152[812], DS152[813], DS152[814], DS152[815], DS152[816], DS152[822], DS152[823], DS152[827], DS152[828], DS152[829], DS152[830], DS152[831], DS152[832], DS152[833], DS152[834], DS152[835], DS152[836], DS152[837], DS152[838], DS152[839], DS152[840], DS152[841], DS152[842], DS152[843], DS152[844], DS152[845], DS152[846], DS152[847], DS152[848], DS152[849], DS152[850], DS152[851], DS152[852], DS152[853], DS152[854], DS152[855], DS152[856], DS152[857], DS152[858], DS152[859], DS152[860], DS152[861], DS152[862], DS152[863], DS152[864], DS152[865], DS152[866], DS152[867], DS152[868], DS152[869], DS152[870], DS152[871], DS152[872], DS152[873], DS152[874], DS152[875], DS152[876], DS152[877], DS152[878], DS152[879], DS152[880]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,152]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,152] is 64 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,152] is RDS152:=[ [ 2, 3, 6, 8, 9, 11, 13, 20, 22, 26, 27, 28, 30, 31, 37, 38, 42, 43, 46, 49, 53, 54, 55, 56, 58, 59, 61, 64 ], [ 2, 3, 6, 8, 9, 11, 13, 20, 22, 23, 26, 27, 28, 30, 31, 37, 38, 42, 44, 45, 46, 49, 53, 54, 55, 56, 60, 61 ], [ 2, 3, 6, 8, 9, 11, 13, 17, 20, 22, 26, 27, 28, 30, 31, 37, 38, 39, 40, 42, 46, 49, 53, 54, 55, 56, 57, 61 ], [ 2, 3, 6, 8, 9, 10, 11, 13, 15, 20, 22, 26, 28, 36, 37, 42, 43, 46, 50, 52, 53, 54, 55, 58, 59, 61, 62, 64 ], [ 2, 3, 6, 8, 9, 10, 11, 13, 15, 20, 22, 23, 26, 28, 36, 37, 42, 44, 45, 46, 50, 52, 53, 54, 55, 60, 61, 62 ], [ 2, 3, 6, 8, 9, 10, 11, 13, 15, 17, 20, 22, 26, 28, 36, 37, 39, 40, 42, 46, 50, 52, 53, 54, 55, 57, 61, 62 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 15, 21, 22, 24, 26, 27, 28, 33, 35, 36, 37, 43, 47, 49, 52, 53, 55, 58, 59, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 15, 21, 22, 23, 24, 26, 27, 28, 33, 35, 36, 37, 44, 45, 47, 49, 52, 53, 55, 60 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 15, 17, 21, 22, 24, 26, 27, 28, 33, 35, 36, 37, 39, 40, 47, 49, 52, 53, 55, 57 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 15, 21, 22, 24, 26, 28, 36, 37, 43, 47, 50, 52, 53, 54, 55, 58, 59, 62, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 15, 21, 22, 23, 24, 26, 28, 36, 37, 44, 45, 47, 50, 52, 53, 54, 55, 60, 62 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 15, 17, 21, 22, 24, 26, 28, 36, 37, 39, 40, 47, 50, 52, 53, 54, 55, 57, 62 ], [ 2, 3, 4, 8, 9, 11, 13, 17, 19, 26, 27, 28, 30, 31, 37, 38, 40, 43, 46, 49, 53, 54, 55, 56, 58, 59, 61, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 17, 19, 23, 26, 27, 28, 30, 31, 37, 38, 40, 44, 45, 46, 49, 53, 54, 55, 56, 60, 61 ], [ 2, 3, 4, 8, 9, 10, 11, 15, 19, 24, 26, 27, 28, 33, 35, 36, 37, 39, 43, 47, 49, 52, 53, 55, 57, 58, 59, 64 ], [ 2, 3, 4, 8, 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