### 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 [64,113]. 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 [64,113]. # 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(64,113));' can be # used to create a list e of the elements of the group [64,113]. # In our list of (64,28,12) DSs in group [64,113], we have [1, # 2, 3, 4, 5, 8, 9, 12, 14, 17, 18, 19, 25, 26, 30, # 39, 40, 43, 45, 48, 49, 50, 51, 52, 55, 58, 60, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[8], e[9], e[12], e[14], e[17], e[18], e[19], e[25], e[26], e[30], e[39], # e[40], e[43], e[45], e[48], e[49], e[50], e[51], e[52], e[55], e[58], e[60], e[63]] # where e is the list of the elements of the group [64,113] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,113));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 113] ### The number of inequivalent (64,28,12) DSs in group [64,113] is 990 ### The list of all inequivalent (64,28,12) DSs in group [64,113] is DS113:=[ [ 1, 2, 3, 4, 5, 8, 9, 12, 14, 17, 18, 19, 25, 26, 30, 39, 40, 43, 45, 48, 49, 50, 51, 52, 55, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 14, 17, 18, 19, 23, 25, 28, 29, 30, 39, 40, 44, 47, 51, 52, 55, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 14, 17, 18, 19, 23, 25, 26, 30, 39, 40, 44, 48, 49, 50, 51, 52, 55, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 14, 15, 16, 17, 23, 25, 28, 29, 30, 36, 37, 44, 47, 51, 52, 55, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 14, 15, 16, 17, 23, 25, 26, 30, 36, 37, 44, 48, 49, 50, 51, 52, 55, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 14, 17, 18, 19, 28, 29, 31, 39, 40, 43, 44, 46, 47, 48, 51, 52, 55, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 14, 17, 18, 19, 23, 28, 29, 31, 39, 40, 45, 46, 47, 48, 51, 52, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 14, 17, 18, 19, 23, 25, 26, 28, 31, 39, 40, 45, 48, 50, 52, 55, 58, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 14, 15, 16, 17, 23, 28, 29, 31, 36, 37, 45, 46, 47, 48, 51, 52, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 14, 15, 16, 17, 23, 25, 26, 28, 31, 36, 37, 45, 48, 50, 52, 55, 58, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 32, 39, 40, 43, 44, 46, 47, 48, 49, 50, 51, 55, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 26, 32, 39, 40, 43, 45, 46, 49, 50, 51, 55, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 26, 28, 32, 39, 40, 43, 44, 45, 46, 50, 55, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 26, 28, 29, 39, 40, 43, 44, 46, 48, 52, 55, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 25, 28, 32, 39, 40, 43, 44, 47, 50, 55, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 25, 26, 28, 39, 40, 43, 45, 50, 51, 52, 55, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 19, 25, 26, 28, 29, 39, 40, 43, 52, 55, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 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5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 21, 34, 37, 38, 39, 42, 43, 45, 49, 53, 57, 58, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 21, 34, 37, 38, 41, 42, 43, 45, 49, 53, 58, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 18, 25, 27, 28, 34, 37, 38, 45, 46, 49, 50, 54, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 18, 25, 27, 28, 30, 34, 37, 38, 44, 45, 50, 52, 54, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 22, 28, 34, 37, 38, 40, 42, 44, 45, 54, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 20, 22, 28, 35, 37, 38, 39, 40, 44, 46, 53, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 19, 20, 28, 37, 39, 42, 44, 46, 53, 54, 56, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 25, 28, 34, 37, 38, 39, 40, 45, 46, 54, 57, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 27, 30, 32, 35, 45, 46, 49, 50, 53, 56, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 22, 28, 36, 40, 42, 44, 46, 54, 55, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 21, 25, 28, 35, 37, 38, 40, 43, 51, 53, 56, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 24, 25, 28, 34, 38, 40, 42, 45, 47, 55, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 21, 25, 37, 38, 39, 43, 51, 54, 56, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 20, 24, 25, 28, 34, 35, 38, 39, 42, 45, 47, 54, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 21, 33, 36, 37, 39, 42, 44, 45, 49, 56, 58, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 22, 25, 28, 35, 36, 40, 42, 45, 46, 54, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 20, 22, 25, 28, 35, 39, 40, 45, 46, 53, 54, 56, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 20, 25, 28, 34, 35, 38, 40, 42, 45, 46, 54, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 27, 32, 36, 37, 44, 45, 50, 52, 58, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 20, 28, 34, 37, 38, 39, 42, 43, 44, 45, 57, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 19, 21, 33, 39, 42, 46, 48, 49, 53, 55, 58, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 20, 22, 28, 37, 39, 40, 44, 45, 53, 56, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 20, 28, 34, 37, 38, 40, 42, 44, 45, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 33, 37, 38, 39, 42, 44, 45, 49, 58, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 19, 20, 28, 38, 40, 42, 44, 46, 55, 57, 59, 60, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 615 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS113[3], DS113[14], DS113[23], DS113[28], DS113[34], DS113[35], DS113[39], DS113[46], DS113[47], DS113[53], DS113[57], DS113[62], DS113[66], DS113[71], DS113[78], DS113[79], DS113[93], DS113[96], DS113[98], DS113[100], DS113[101], DS113[103], DS113[107], DS113[114], DS113[115], DS113[120], DS113[123], DS113[127], DS113[128], DS113[130], DS113[132], DS113[133], DS113[136], DS113[140], DS113[141], DS113[145], DS113[146], DS113[151], DS113[157], DS113[159], DS113[160], DS113[163], DS113[166], DS113[185], DS113[189], DS113[190], DS113[194], DS113[195], DS113[196], DS113[205], DS113[206], DS113[207], DS113[209], DS113[211], DS113[212], DS113[217], DS113[223], DS113[224], DS113[229], DS113[230], DS113[234], DS113[235], DS113[236], DS113[238], DS113[244], DS113[245], DS113[246], DS113[247], DS113[248], DS113[254], DS113[261], DS113[264], DS113[265], DS113[266], DS113[268], DS113[270], DS113[272], DS113[273], DS113[275], DS113[277], DS113[281], DS113[283], DS113[284], DS113[285], DS113[286], DS113[289], DS113[292], DS113[295], DS113[296], DS113[298], DS113[304], DS113[305], DS113[306], DS113[309], DS113[310], DS113[311], DS113[314], DS113[316], DS113[317], DS113[319], DS113[323], DS113[328], DS113[331], DS113[337], DS113[338], DS113[339], DS113[347], DS113[348], DS113[360], DS113[361], DS113[362], DS113[363], DS113[368], DS113[369], DS113[370], DS113[371], DS113[375], DS113[376], DS113[384], DS113[389], DS113[390], DS113[391], DS113[394], DS113[395], DS113[396], DS113[397], DS113[399], DS113[401], DS113[402], DS113[403], DS113[404], DS113[406], DS113[408], DS113[415], DS113[418], DS113[422], DS113[423], DS113[426], DS113[430], DS113[432], DS113[437], DS113[441], DS113[442], DS113[447], DS113[449], DS113[454], DS113[455], DS113[461], DS113[462], DS113[463], DS113[464], DS113[465], DS113[466], DS113[467], DS113[468], DS113[469], DS113[470], DS113[471], DS113[472], DS113[475], DS113[476], DS113[477], DS113[478], DS113[479], DS113[480], DS113[481], DS113[482], DS113[484], DS113[486], DS113[487], DS113[488], DS113[489], DS113[490], DS113[491], DS113[493], DS113[494], DS113[495], DS113[500], DS113[501], DS113[504], DS113[505], DS113[506], DS113[509], DS113[510], DS113[511], DS113[512], DS113[513], DS113[514], DS113[515], DS113[516], DS113[517], DS113[518], DS113[519], DS113[520], DS113[521], DS113[522], DS113[523], DS113[524], DS113[525], DS113[526], DS113[527], DS113[528], DS113[529], DS113[530], DS113[531], DS113[532], DS113[533], DS113[534], DS113[535], DS113[536], DS113[537], DS113[538], DS113[539], DS113[540], DS113[541], DS113[542], DS113[543], DS113[544], DS113[545], DS113[546], DS113[547], DS113[548], DS113[549], DS113[550], DS113[553], DS113[554], DS113[555], DS113[556], DS113[557], DS113[559], DS113[561], DS113[563], DS113[564], DS113[565], DS113[566], DS113[567], DS113[568], DS113[569], DS113[570], DS113[571], DS113[572], DS113[573], DS113[574], DS113[575], DS113[576], DS113[577], DS113[578], DS113[579], DS113[580], DS113[581], DS113[582], DS113[583], DS113[584], DS113[585], DS113[586], DS113[587], DS113[588], DS113[589], DS113[590], DS113[591], DS113[592], DS113[593], DS113[594], DS113[595], DS113[596], DS113[597], DS113[598], DS113[599], DS113[600], DS113[601], DS113[602], DS113[603], DS113[604], DS113[605], DS113[606], DS113[607], DS113[608], DS113[609], DS113[611], DS113[613], DS113[616], DS113[617], DS113[618], DS113[619], DS113[620], DS113[621], DS113[622], DS113[623], DS113[624], DS113[625], DS113[626], DS113[627], DS113[628], DS113[629], DS113[630], DS113[631], DS113[632], DS113[633], DS113[634], DS113[635], DS113[636], DS113[649], DS113[650], DS113[651], DS113[652], DS113[653], DS113[654], DS113[655], DS113[656], DS113[657], DS113[658], DS113[659], DS113[660], DS113[664], DS113[665], DS113[666], DS113[667], DS113[668], DS113[669], DS113[670], DS113[671], DS113[675], DS113[676], DS113[677], DS113[678], DS113[679], DS113[680], DS113[681], DS113[682], DS113[683], DS113[684], DS113[685], DS113[686], DS113[687], DS113[688], DS113[689], DS113[690], DS113[691], DS113[692], DS113[693], DS113[694], DS113[695], DS113[696], DS113[697], DS113[698], DS113[699], DS113[700], DS113[701], DS113[702], DS113[703], DS113[704], DS113[705], DS113[706], DS113[707], DS113[708], DS113[709], DS113[710], DS113[711], DS113[712], DS113[713], DS113[714], DS113[715], DS113[716], DS113[717], DS113[718], DS113[719], DS113[720], DS113[723], DS113[724], DS113[725], DS113[726], DS113[727], DS113[728], DS113[729], DS113[730], DS113[731], DS113[732], DS113[733], DS113[734], DS113[735], DS113[736], DS113[737], DS113[738], DS113[739], DS113[740], DS113[741], DS113[742], DS113[743], DS113[744], DS113[745], DS113[746], DS113[747], DS113[748], DS113[749], DS113[750], DS113[751], DS113[752], DS113[753], DS113[754], DS113[755], DS113[756], DS113[757], DS113[758], DS113[759], DS113[760], DS113[761], DS113[762], DS113[763], DS113[764], DS113[765], DS113[766], DS113[767], DS113[768], DS113[769], DS113[770], DS113[771], DS113[772], DS113[773], DS113[774], DS113[775], DS113[776], DS113[777], DS113[778], DS113[779], DS113[780], DS113[781], DS113[782], DS113[783], DS113[784], DS113[785], DS113[786], DS113[787], DS113[788], DS113[789], DS113[790], DS113[791], DS113[792], DS113[793], DS113[794], DS113[795], DS113[796], DS113[797], DS113[798], DS113[799], DS113[800], DS113[801], DS113[802], DS113[803], DS113[804], DS113[805], DS113[806], DS113[807], DS113[808], DS113[809], DS113[810], DS113[811], DS113[812], DS113[813], DS113[814], DS113[815], DS113[816], DS113[817], DS113[818], DS113[819], DS113[822], DS113[824], DS113[827], DS113[830], DS113[831], DS113[832], DS113[833], DS113[834], DS113[835], DS113[836], DS113[837], DS113[838], DS113[839], DS113[840], DS113[841], DS113[842], DS113[843], DS113[844], DS113[846], DS113[852], DS113[856], DS113[857], DS113[858], DS113[859], DS113[863], DS113[865], DS113[866], DS113[867], DS113[868], DS113[869], DS113[870], DS113[871], DS113[872], DS113[873], DS113[874], DS113[875], DS113[876], DS113[877], DS113[878], DS113[879], DS113[880], DS113[881], DS113[882], DS113[883], DS113[884], DS113[885], DS113[886], DS113[887], DS113[888], DS113[889], DS113[890], DS113[891], DS113[892], DS113[893], DS113[894], DS113[895], DS113[896], DS113[897], DS113[898], DS113[899], DS113[900], DS113[901], DS113[902], DS113[903], DS113[904], DS113[905], DS113[906], DS113[907], DS113[908], DS113[909], DS113[910], DS113[911], DS113[912], DS113[913], DS113[914], DS113[915], DS113[916], DS113[917], DS113[918], DS113[919], DS113[920], DS113[921], DS113[922], DS113[923], DS113[924], DS113[925], DS113[926], DS113[927], DS113[928], DS113[929], DS113[930], DS113[931], DS113[932], DS113[933], DS113[934], DS113[935], DS113[936], DS113[937], DS113[938], DS113[939], DS113[940], DS113[941], DS113[942], DS113[943], DS113[944], DS113[945], DS113[946], DS113[947], DS113[948], DS113[949], DS113[950], DS113[951], DS113[952], DS113[953], DS113[954], DS113[955], DS113[956], DS113[957], DS113[958], DS113[959], DS113[960], DS113[961], DS113[962], DS113[963], DS113[964], DS113[965], DS113[966], DS113[967], DS113[968], DS113[969], DS113[970], DS113[971], DS113[972], DS113[973], DS113[974], DS113[975], DS113[976], DS113[977], DS113[978], DS113[979], DS113[980], DS113[981], DS113[982], DS113[983], DS113[984], DS113[985], DS113[986], DS113[987], DS113[988], DS113[989], DS113[990]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,113]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,113] is 40 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,113] is RDS113:=[ [ 2, 3, 5, 7, 8, 9, 10, 15, 18, 23, 26, 27, 28, 32, 34, 36, 37, 38, 44, 45, 46, 49, 52, 56, 57, 60, 61, 63 ], [ 2, 3, 5, 7, 8, 9, 10, 13, 18, 23, 26, 27, 28, 32, 33, 34, 35, 37, 44, 45, 46, 49, 52, 54, 57, 60, 61, 63 ], [ 2, 3, 5, 6, 8, 9, 10, 15, 16, 21, 22, 23, 25, 27, 28, 32, 34, 38, 44, 47, 49, 52, 53, 55, 59, 61, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 15, 16, 18, 21, 22, 23, 25, 27, 28, 32, 38, 39, 41, 44, 47, 49, 52, 57, 59, 61, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 13, 15, 16, 21, 22, 23, 25, 27, 28, 32, 33, 35, 38, 44, 47, 49, 52, 54, 59, 61, 64 ], [ 2, 3, 5, 6, 7, 8, 9, 10, 18, 20, 22, 23, 26, 27, 28, 32, 34, 37, 42, 44, 45, 46, 49, 52, 57, 60, 61, 63 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 21, 26, 32, 33, 34, 43, 45, 46, 49, 50, 51, 53, 55, 58, 60, 61, 63 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 21, 23, 26, 32, 33, 34, 44, 46, 49, 50, 51, 53, 55, 59, 61, 63, 64 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 21, 23, 25, 29, 32, 33, 34, 44, 47, 49, 53, 55, 59, 61, 62, 63, 64 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 18, 21, 26, 32, 33, 39, 41, 43, 45, 46, 49, 50, 51, 57, 58, 60, 61 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 18, 21, 23, 26, 32, 33, 39, 41, 44, 46, 49, 50, 51, 57, 59, 61, 64 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 17, 18, 21, 23, 25, 29, 32, 33, 39, 41, 44, 47, 49, 57, 59, 61, 62, 64 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17, 21, 26, 32, 33, 36, 38, 43, 45, 46, 49, 50, 51, 56, 58, 60, 61 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17, 21, 23, 26, 32, 33, 36, 38, 44, 46, 49, 50, 51, 56, 59, 61, 64 ], [ 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17, 21, 23, 25, 29, 32, 33, 36, 38, 44, 47, 49, 56, 59, 61, 62, 64 ], [ 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 17, 20, 21, 22, 26, 32, 33, 42, 43, 45, 46, 49, 50, 51, 58, 60, 61 ], [ 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 17, 20, 21, 22, 23, 26, 32, 33, 42, 44, 46, 49, 50, 51, 59, 61, 64 ], [ 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 17, 20, 21, 22, 23, 25, 29, 32, 33, 42, 44, 47, 49, 59, 61, 62, 64 ], [ 2, 3, 4, 5, 7, 8, 9, 10, 17, 18, 19, 23, 26, 27, 28, 32, 34, 37, 40, 44, 45, 46, 49, 52, 57, 60, 61, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 32, 38, 40, 44, 47, 49, 52, 59, 61, 64 ], [ 1, 2, 3, 8, 9, 10, 15, 18, 21, 23, 26, 27, 28, 32, 36, 37, 38, 44, 45, 46, 49, 52, 53, 55, 56, 57, 60, 61 ], [ 1, 2, 3, 8, 9, 10, 13, 18, 21, 23, 26, 27, 28, 32, 33, 35, 37, 44, 45, 46, 49, 52, 53, 54, 55, 57, 60, 61 ], [ 1, 2, 3, 6, 8, 9, 10, 18, 20, 21, 22, 23, 26, 27, 28, 32, 37, 42, 44, 45, 46, 49, 52, 53, 55, 57, 60, 61 ], [ 1, 2, 3, 6, 7, 8, 9, 10, 16, 22, 23, 25, 27, 28, 32, 34, 36, 44, 47, 49, 52, 53, 55, 56, 59, 61, 63, 64 ], [ 1, 2, 3, 6, 7, 8, 9, 10, 16, 18, 22, 23, 25, 27, 28, 32, 36, 39, 41, 44, 47, 49, 52, 56, 57, 59, 61, 64 ], [ 1, 2, 3, 6, 7, 8, 9, 10, 13, 16, 22, 23, 25, 27, 28, 32, 33, 35, 36, 44, 47, 49, 52, 54, 56, 59, 61, 64 ], [ 1, 2, 3, 4, 8, 9, 10, 17, 18, 19, 21, 23, 26, 27, 28, 32, 37, 40, 44, 45, 46, 49, 52, 53, 55, 57, 60, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 10, 16, 17, 19, 22, 23, 25, 27, 28, 32, 36, 40, 44, 47, 49, 52, 56, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 26, 32, 34, 35, 43, 45, 46, 49, 50, 51, 53, 54, 55, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 23, 26, 32, 34, 35, 44, 46, 49, 50, 51, 53, 54, 55, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 23, 25, 29, 32, 34, 35, 44, 47, 49, 53, 54, 55, 59, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 26, 32, 35, 39, 41, 43, 45, 46, 49, 50, 51, 54, 57, 58, 60, 61 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 23, 26, 32, 35, 39, 41, 44, 46, 49, 50, 51, 54, 57, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 17, 18, 23, 25, 29, 32, 35, 39, 41, 44, 47, 49, 54, 57, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 15, 17, 26, 32, 35, 36, 38, 43, 45, 46, 49, 50, 51, 54, 56, 58, 60, 61 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 15, 17, 23, 26, 32, 35, 36, 38, 44, 46, 49, 50, 51, 54, 56, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 11, 12, 14, 15, 17, 23, 25, 29, 32, 35, 36, 38, 44, 47, 49, 54, 56, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20, 22, 26, 32, 35, 42, 43, 45, 46, 49, 50, 51, 54, 58, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20, 22, 23, 26, 32, 35, 42, 44, 46, 49, 50, 51, 54, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20, 22, 23, 25, 29, 32, 35, 42, 44, 47, 49, 54, 59, 61, 62, 64 ] ]; ########################### ########################### ###########################