### 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 [64,178]. 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 [64,178]. # 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(64,178));' can be # used to create a list e of the elements of the group [64,178]. # In our list of (64,28,12) DSs in group [64,178], we have [1, # 2, 3, 4, 5, 9, 10, 12, 13, 14, 18, 19, 20, 22, 23, # 29, 31, 33, 34, 38, 41, 43, 44, 49, 53, 56, 62, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[9], e[10], e[12], e[13], e[14], e[18], e[19], e[20], e[22], e[23], e[29], # e[31], e[33], e[34], e[38], e[41], e[43], e[44], e[49], e[53], e[56], e[62], e[64]] # where e is the list of the elements of the group [64,178] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,178));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 178] ### The number of inequivalent (64,28,12) DSs in group [64,178] is 952 ### The list of all inequivalent (64,28,12) DSs in group [64,178] is DS178:=[ [ 1, 2, 3, 4, 5, 9, 10, 12, 13, 14, 18, 19, 20, 22, 23, 29, 31, 33, 34, 38, 41, 43, 44, 49, 53, 56, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 19, 21, 23, 24, 25, 27, 32, 39, 43, 46, 49, 51, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 19, 20, 23, 27, 29, 30, 33, 38, 41, 43, 44, 50, 52, 55, 56, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 23, 29, 31, 33, 38, 41, 43, 44, 49, 55, 56, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 16, 17, 19, 20, 23, 25, 26, 30, 31, 32, 33, 34, 36, 40, 43, 48, 56, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 16, 19, 20, 25, 26, 30, 31, 32, 33, 34, 37, 39, 43, 44, 45, 48, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 16, 19, 20, 23, 25, 26, 30, 31, 32, 33, 34, 37, 39, 43, 48, 57, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 13, 16, 17, 19, 23, 25, 26, 31, 36, 40, 42, 48, 52, 53, 54, 55, 56, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 13, 16, 17, 19, 23, 25, 26, 31, 36, 40, 42, 43, 48, 52, 53, 54, 55, 56, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 13, 14, 16, 19, 23, 25, 26, 31, 37, 39, 42, 43, 48, 52, 53, 54, 55, 57, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 16, 17, 19, 20, 23, 24, 32, 33, 36, 40, 44, 48, 55, 56, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 20, 23, 24, 32, 35, 37, 39, 40, 41, 44, 48, 53, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 19, 20, 23, 24, 32, 33, 37, 39, 44, 48, 55, 57, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 18, 20, 23, 26, 32, 33, 37, 40, 43, 46, 55, 57, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 18, 20, 23, 24, 32, 35, 37, 40, 43, 48, 53, 57, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 18, 20, 23, 24, 32, 33, 37, 40, 45, 48, 55, 57, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 16, 17, 20, 23, 24, 32, 35, 37, 41, 44, 48, 53, 57, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 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36, 40, 48, 49, 50, 51, 54, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 18, 20, 25, 26, 30, 31, 32, 35, 37, 40, 48, 49, 50, 51, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 28, 30, 36, 37, 38, 41, 43, 45, 48, 50, 52, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 27, 29, 30, 36, 37, 38, 41, 44, 48, 50, 52, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 27, 29, 30, 36, 37, 38, 41, 43, 44, 48, 50, 52, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 27, 28, 30, 36, 37, 38, 41, 43, 45, 48, 49, 52, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 26, 27, 30, 36, 37, 38, 41, 44, 47, 51, 52, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 26, 27, 30, 36, 37, 38, 41, 43, 44, 47, 51, 52, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 26, 27, 28, 30, 36, 37, 38, 41, 47, 49, 52, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 18, 19, 20, 23, 24, 26, 27, 28, 30, 36, 37, 38, 41, 43, 47, 49, 52, 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58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 27, 29, 37, 38, 39, 40, 46, 49, 52, 55, 57, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 27, 29, 34, 37, 38, 39, 40, 46, 52, 57, 58, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 20, 26, 35, 36, 38, 40, 42, 44, 45, 49, 50, 53, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 26, 37, 38, 40, 42, 45, 49, 50, 53, 55, 56, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20, 26, 34, 37, 38, 40, 42, 44, 45, 49, 50, 53, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 30, 32, 37, 38, 39, 40, 46, 54, 55, 57, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 23, 26, 36, 38, 40, 42, 44, 49, 50, 53, 55, 62, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 457 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS178[2], DS178[4], DS178[10], DS178[13], DS178[19], DS178[22], DS178[25], DS178[27], DS178[28], DS178[29], DS178[33], DS178[34], DS178[35], DS178[38], DS178[40], DS178[43], DS178[48], DS178[49], DS178[50], DS178[51], DS178[52], DS178[53], DS178[54], DS178[60], DS178[61], DS178[62], DS178[64], DS178[69], DS178[77], DS178[86], DS178[87], DS178[96], DS178[102], DS178[103], DS178[104], DS178[106], DS178[108], DS178[110], DS178[111], DS178[112], DS178[114], DS178[118], DS178[119], DS178[120], DS178[123], DS178[125], DS178[132], DS178[133], DS178[134], DS178[135], DS178[136], DS178[143], DS178[146], DS178[147], DS178[148], DS178[154], DS178[155], DS178[156], DS178[159], DS178[165], DS178[166], DS178[167], DS178[169], DS178[170], DS178[171], DS178[176], DS178[177], DS178[179], DS178[180], DS178[181], DS178[183], DS178[188], DS178[190], DS178[193], DS178[195], DS178[196], DS178[197], DS178[203], DS178[205], DS178[207], DS178[208], DS178[209], DS178[212], DS178[215], DS178[217], DS178[218], DS178[225], DS178[231], DS178[234], DS178[241], DS178[248], DS178[251], DS178[253], DS178[258], DS178[260], DS178[261], DS178[266], DS178[268], DS178[269], DS178[274], DS178[275], DS178[276], DS178[279], DS178[285], DS178[288], DS178[290], DS178[292], DS178[294], DS178[297], DS178[302], DS178[305], DS178[306], DS178[308], DS178[315], DS178[318], DS178[321], DS178[327], DS178[330], DS178[332], DS178[336], DS178[337], DS178[338], DS178[340], DS178[341], DS178[345], DS178[354], DS178[362], DS178[363], DS178[364], DS178[366], DS178[367], DS178[368], DS178[369], DS178[370], DS178[371], DS178[374], DS178[375], DS178[376], DS178[380], DS178[384], DS178[385], DS178[386], DS178[387], DS178[388], DS178[389], DS178[395], DS178[396], DS178[402], DS178[403], DS178[414], DS178[416], DS178[417], DS178[419], DS178[421], DS178[422], DS178[423], DS178[428], DS178[435], DS178[436], DS178[438], DS178[446], DS178[450], DS178[451], DS178[452], DS178[453], DS178[454], DS178[455], DS178[456], DS178[463], DS178[464], DS178[465], DS178[466], DS178[467], DS178[468], DS178[469], DS178[470], DS178[474], DS178[476], DS178[480], DS178[482], DS178[484], DS178[485], DS178[489], DS178[491], DS178[493], DS178[494], DS178[495], DS178[498], DS178[500], DS178[507], DS178[508], DS178[509], DS178[512], DS178[516], DS178[518], DS178[520], DS178[522], DS178[525], DS178[526], DS178[527], DS178[529], DS178[532], DS178[539], DS178[540], DS178[541], DS178[542], DS178[544], DS178[545], DS178[546], DS178[547], DS178[548], DS178[551], DS178[558], DS178[561], DS178[562], DS178[565], DS178[566], DS178[567], DS178[569], DS178[571], DS178[572], DS178[573], DS178[574], DS178[576], DS178[582], DS178[583], DS178[585], DS178[602], DS178[603], DS178[605], DS178[606], DS178[610], DS178[614], DS178[617], DS178[618], DS178[619], DS178[620], DS178[621], DS178[622], DS178[623], DS178[624], DS178[627], DS178[633], DS178[635], DS178[636], DS178[648], DS178[654], DS178[655], DS178[656], DS178[657], DS178[658], DS178[659], DS178[660], DS178[661], DS178[662], DS178[663], DS178[664], DS178[683], DS178[685], DS178[692], DS178[695], DS178[696], DS178[701], DS178[704], DS178[712], DS178[714], DS178[715], DS178[716], DS178[717], DS178[723], DS178[725], DS178[726], DS178[731], DS178[732], DS178[733], DS178[734], DS178[736], DS178[737], DS178[739], DS178[740], DS178[741], DS178[742], DS178[744], DS178[745], DS178[746], DS178[747], DS178[748], DS178[749], DS178[750], DS178[751], DS178[753], DS178[754], DS178[755], DS178[756], DS178[758], DS178[759], DS178[760], DS178[761], DS178[763], DS178[766], DS178[767], DS178[768], DS178[769], DS178[770], DS178[771], DS178[772], DS178[773], DS178[774], DS178[775], DS178[776], DS178[777], DS178[778], DS178[779], DS178[780], DS178[783], DS178[784], DS178[788], DS178[789], DS178[790], DS178[791], DS178[792], DS178[795], DS178[796], DS178[797], DS178[798], DS178[799], DS178[800], DS178[801], DS178[802], DS178[803], DS178[805], DS178[807], DS178[808], DS178[809], DS178[810], DS178[812], DS178[815], DS178[816], DS178[817], DS178[820], DS178[821], DS178[822], DS178[823], DS178[824], DS178[825], DS178[826], DS178[828], DS178[829], DS178[831], DS178[832], DS178[833], DS178[838], DS178[839], DS178[841], DS178[843], DS178[845], DS178[847], DS178[849], DS178[850], DS178[851], DS178[852], DS178[853], DS178[854], DS178[855], DS178[856], DS178[857], DS178[858], DS178[859], DS178[860], DS178[861], DS178[862], DS178[863], DS178[864], DS178[865], DS178[866], DS178[867], DS178[868], DS178[869], DS178[870], DS178[871], DS178[872], DS178[873], DS178[874], DS178[875], DS178[876], DS178[877], DS178[878], DS178[879], DS178[880], DS178[881], DS178[882], DS178[884], DS178[885], DS178[886], DS178[887], DS178[888], DS178[889], DS178[890], DS178[891], DS178[892], DS178[893], DS178[894], DS178[895], DS178[896], DS178[897], DS178[898], DS178[899], DS178[900], DS178[901], DS178[902], DS178[903], DS178[904], DS178[905], DS178[906], DS178[907], DS178[908], DS178[909], DS178[910], DS178[911], DS178[912], DS178[913], DS178[914], DS178[915], DS178[916], DS178[917], DS178[918], DS178[920], DS178[921], DS178[922], DS178[923], DS178[924], DS178[925], DS178[926], DS178[927], DS178[928], DS178[929], DS178[930], DS178[931], DS178[932], DS178[933], DS178[934], DS178[935], DS178[936], DS178[937], DS178[938], DS178[939], DS178[940], DS178[943], DS178[944], DS178[945], DS178[946], DS178[947], DS178[948], DS178[949], DS178[950], DS178[951], DS178[952]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,178]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,178] is 64 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,178] is RDS178:=[ [ 2, 3, 4, 8, 9, 12, 13, 15, 17, 18, 23, 24, 27, 29, 30, 37, 44, 45, 46, 48, 50, 52, 53, 54, 55, 56, 57, 60 ], [ 2, 3, 4, 8, 9, 12, 13, 15, 17, 18, 23, 24, 25, 27, 29, 30, 37, 44, 45, 50, 52, 53, 54, 55, 56, 57, 60, 61 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 15, 17, 18, 25, 27, 29, 32, 37, 43, 44, 46, 47, 50, 53, 54, 55, 56, 57, 59, 60 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 15, 17, 18, 24, 25, 27, 29, 32, 37, 43, 44, 50, 53, 54, 55, 56, 57, 59, 60, 61 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 15, 17, 18, 23, 25, 27, 29, 32, 37, 45, 46, 47, 50, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 8, 9, 11, 12, 13, 15, 17, 18, 23, 24, 25, 27, 29, 32, 37, 45, 50, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 29, 31, 37, 43, 47, 48, 49, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 24, 29, 31, 37, 43, 46, 48, 49, 53, 54, 55, 56, 57, 58, 59, 62, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 24, 25, 29, 31, 37, 43, 49, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 29, 31, 37, 44, 45, 47, 48, 49, 53, 54, 55, 56, 57, 60, 61, 62 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 28, 29, 31, 37, 45, 47, 48, 51, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 25, 28, 29, 31, 37, 45, 46, 47, 51, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 24, 29, 31, 37, 44, 45, 46, 48, 49, 53, 54, 55, 56, 57, 60, 62 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 24, 28, 29, 31, 37, 45, 46, 48, 51, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 24, 25, 29, 31, 37, 44, 45, 49, 53, 54, 55, 56, 57, 60, 61, 62 ], [ 2, 3, 4, 8, 9, 10, 12, 13, 15, 17, 18, 23, 24, 25, 28, 29, 31, 37, 45, 51, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 6, 8, 12, 13, 15, 17, 18, 20, 22, 23, 24, 30, 37, 42, 44, 45, 46, 48, 52, 53, 54, 55, 56, 57, 60 ], [ 2, 3, 4, 6, 8, 12, 13, 15, 17, 18, 20, 22, 23, 24, 25, 30, 37, 42, 44, 45, 52, 53, 54, 55, 56, 57, 60, 61 ], [ 2, 3, 4, 6, 8, 11, 12, 13, 15, 17, 18, 20, 22, 23, 25, 32, 37, 42, 45, 46, 47, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 6, 8, 11, 12, 13, 15, 17, 18, 20, 22, 23, 24, 25, 32, 37, 42, 45, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 5, 8, 12, 13, 15, 17, 18, 20, 21, 23, 24, 30, 37, 42, 43, 45, 46, 48, 52, 53, 54, 55, 56, 57, 59 ], [ 2, 3, 4, 5, 8, 12, 13, 15, 17, 18, 20, 21, 23, 24, 25, 30, 37, 42, 43, 45, 52, 53, 54, 55, 56, 57, 59, 61 ], [ 2, 3, 4, 5, 8, 10, 12, 13, 15, 17, 18, 20, 21, 23, 31, 37, 42, 45, 47, 48, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 5, 8, 10, 12, 13, 15, 17, 18, 20, 21, 23, 25, 31, 37, 42, 45, 46, 47, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 5, 8, 10, 12, 13, 15, 17, 18, 20, 21, 23, 24, 31, 37, 42, 45, 46, 48, 53, 54, 55, 56, 57, 58, 64 ], [ 2, 3, 4, 5, 8, 10, 12, 13, 15, 17, 18, 20, 21, 23, 24, 25, 31, 37, 42, 45, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 2, 3, 4, 5, 8, 9, 12, 13, 15, 17, 18, 20, 21, 24, 25, 27, 29, 30, 37, 42, 50, 52, 53, 54, 55, 56, 57, 61 ], [ 2, 3, 4, 5, 8, 9, 10, 12, 13, 15, 17, 18, 20, 21, 29, 31, 37, 42, 47, 48, 49, 53, 54, 55, 56, 57, 61, 62 ], [ 2, 3, 4, 5, 8, 9, 10, 12, 13, 15, 17, 18, 20, 21, 24, 29, 31, 37, 42, 46, 48, 49, 53, 54, 55, 56, 57, 62 ], [ 2, 3, 4, 5, 8, 9, 10, 12, 13, 15, 17, 18, 20, 21, 24, 25, 29, 31, 37, 42, 49, 53, 54, 55, 56, 57, 61, 62 ], [ 2, 3, 4, 5, 6, 8, 11, 12, 13, 15, 17, 18, 21, 22, 23, 24, 32, 37, 43, 45, 46, 48, 53, 54, 55, 56, 57, 59 ], [ 2, 3, 4, 5, 6, 8, 11, 12, 13, 15, 17, 18, 21, 22, 23, 24, 25, 32, 37, 43, 45, 53, 54, 55, 56, 57, 59, 61 ], [ 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 18, 21, 22, 23, 31, 37, 44, 45, 47, 48, 53, 54, 55, 56, 57, 60, 61 ], [ 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 18, 21, 22, 23, 25, 31, 37, 44, 45, 46, 47, 53, 54, 55, 56, 57, 60 ], [ 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 18, 21, 22, 23, 24, 31, 37, 44, 45, 46, 48, 53, 54, 55, 56, 57, 60 ], [ 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 18, 21, 22, 23, 24, 25, 31, 37, 44, 45, 53, 54, 55, 56, 57, 60, 61 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 21, 22, 25, 27, 29, 32, 37, 46, 47, 50, 53, 54, 55, 56, 57 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 21, 22, 24, 25, 27, 29, 32, 37, 50, 53, 54, 55, 56, 57, 61 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 21, 22, 24, 28, 29, 31, 37, 46, 48, 51, 53, 54, 55, 56, 57 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 21, 22, 24, 25, 28, 29, 31, 37, 51, 53, 54, 55, 56, 57, 61 ], [ 1, 2, 3, 4, 7, 8, 11, 12, 13, 15, 17, 18, 20, 23, 24, 32, 37, 42, 43, 45, 46, 48, 53, 54, 55, 56, 57, 59 ], [ 1, 2, 3, 4, 7, 8, 11, 12, 13, 15, 17, 18, 20, 23, 24, 25, 32, 37, 42, 43, 45, 53, 54, 55, 56, 57, 59, 61 ], [ 1, 2, 3, 4, 7, 8, 10, 12, 13, 15, 17, 18, 20, 23, 31, 37, 42, 44, 45, 47, 48, 53, 54, 55, 56, 57, 60, 61 ], [ 1, 2, 3, 4, 7, 8, 10, 12, 13, 15, 17, 18, 20, 23, 25, 31, 37, 42, 44, 45, 46, 47, 53, 54, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 7, 8, 10, 12, 13, 15, 17, 18, 20, 23, 24, 31, 37, 42, 44, 45, 46, 48, 53, 54, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 7, 8, 10, 12, 13, 15, 17, 18, 20, 23, 24, 25, 31, 37, 42, 44, 45, 53, 54, 55, 56, 57, 60, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 12, 13, 15, 17, 18, 22, 23, 24, 30, 37, 43, 45, 46, 48, 52, 53, 54, 55, 56, 57, 59 ], [ 1, 2, 3, 4, 6, 7, 8, 12, 13, 15, 17, 18, 22, 23, 24, 25, 30, 37, 43, 45, 52, 53, 54, 55, 56, 57, 59, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 31, 37, 45, 47, 48, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 25, 31, 37, 45, 46, 47, 53, 54, 55, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 24, 31, 37, 45, 46, 48, 53, 54, 55, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 24, 25, 31, 37, 45, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 15, 17, 18, 22, 24, 25, 27, 29, 30, 37, 50, 52, 53, 54, 55, 56, 57, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 22, 29, 31, 37, 47, 48, 49, 53, 54, 55, 56, 57, 61, 62 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 22, 24, 29, 31, 37, 46, 48, 49, 53, 54, 55, 56, 57, 62 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 22, 24, 25, 29, 31, 37, 49, 53, 54, 55, 56, 57, 61, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 12, 13, 15, 17, 18, 21, 23, 24, 30, 37, 44, 45, 46, 48, 52, 53, 54, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 7, 8, 12, 13, 15, 17, 18, 21, 23, 24, 25, 30, 37, 44, 45, 52, 53, 54, 55, 56, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 15, 17, 18, 21, 23, 25, 32, 37, 45, 46, 47, 53, 54, 55, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 11, 12, 13, 15, 17, 18, 21, 23, 24, 25, 32, 37, 45, 53, 54, 55, 56, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 17, 18, 21, 24, 28, 29, 30, 37, 46, 48, 51, 52, 53, 54, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 17, 18, 21, 24, 25, 28, 29, 30, 37, 51, 52, 53, 54, 55, 56, 57, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 21, 25, 29, 32, 37, 46, 47, 49, 53, 54, 55, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 21, 24, 25, 29, 32, 37, 49, 53, 54, 55, 56, 57, 61, 62 ] ]; ########################### ########################### ###########################