### 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 [64,164]. 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 [64,164]. # 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(64,164));' can be # used to create a list e of the elements of the group [64,164]. # In our list of (64,28,12) DSs in group [64,164], we have [1, # 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 21, 23, 29, # 30, 34, 36, 39, 40, 41, 44, 45, 49, 54, 60, 62, 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[15], e[21], e[23], e[29], e[30], # e[34], e[36], e[39], e[40], e[41], e[44], e[45], e[49], e[54], e[60], e[62], e[63]] # where e is the list of the elements of the group [64,164] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,164));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 164] ### The number of inequivalent (64,28,12) DSs in group [64,164] is 1558 ### The list of all inequivalent (64,28,12) DSs in group [64,164] is DS164:=[ [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 21, 23, 29, 30, 34, 36, 39, 40, 41, 44, 45, 49, 54, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 21, 23, 28, 29, 30, 34, 36, 39, 40, 41, 45, 51, 54, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 19, 21, 23, 30, 34, 35, 36, 39, 44, 49, 50, 51, 55, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 19, 21, 23, 28, 30, 34, 35, 36, 39, 50, 55, 57, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 19, 21, 23, 28, 29, 30, 34, 36, 39, 51, 54, 57, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 19, 21, 23, 28, 29, 30, 34, 36, 39, 44, 51, 54, 57, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 18, 21, 23, 30, 34, 35, 36, 40, 45, 49, 50, 51, 55, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 18, 21, 23, 29, 30, 36, 40, 44, 45, 49, 53, 54, 55, 57, 60, 62 ], [ 1, 2, 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9, 10, 11, 12, 13, 14, 18, 30, 34, 35, 36, 37, 41, 44, 46, 52, 54, 57, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 19, 30, 34, 36, 37, 40, 44, 46, 52, 55, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 30, 34, 35, 37, 39, 41, 44, 51, 52, 54, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 25, 30, 34, 37, 40, 41, 44, 45, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 25, 30, 34, 37, 39, 40, 44, 51, 52, 55, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 25, 28, 30, 34, 37, 40, 52, 53, 55, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 25, 30, 35, 37, 41, 45, 49, 52, 53, 54, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 25, 30, 35, 37, 39, 41, 44, 45, 52, 54, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 25, 28, 30, 35, 37, 39, 52, 54, 55, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 25, 30, 37, 40, 45, 49, 52, 53, 55, 60, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 955 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS164[7], DS164[9], DS164[12], DS164[13], DS164[17], DS164[18], DS164[20], DS164[29], DS164[30], DS164[35], DS164[36], DS164[37], DS164[38], DS164[41], DS164[42], DS164[43], DS164[44], DS164[45], DS164[56], DS164[57], DS164[60], DS164[63], DS164[64], DS164[68], DS164[69], DS164[72], DS164[73], DS164[75], DS164[76], DS164[77], DS164[79], DS164[80], DS164[83], DS164[87], DS164[89], DS164[90], DS164[98], DS164[107], DS164[111], DS164[112], DS164[118], DS164[121], DS164[129], DS164[133], DS164[141], DS164[142], DS164[143], DS164[144], DS164[145], DS164[147], DS164[151], DS164[152], DS164[153], DS164[156], DS164[157], DS164[160], DS164[162], DS164[163], DS164[164], DS164[165], DS164[166], DS164[167], DS164[168], DS164[169], DS164[170], DS164[171], DS164[172], DS164[174], DS164[175], DS164[178], DS164[179], DS164[180], DS164[181], DS164[182], DS164[184], DS164[186], DS164[187], DS164[188], DS164[192], DS164[193], DS164[194], DS164[196], DS164[200], DS164[201], DS164[206], DS164[207], DS164[208], DS164[209], DS164[211], DS164[214], DS164[219], DS164[223], DS164[229], DS164[234], DS164[241], DS164[248], DS164[251], DS164[255], DS164[257], DS164[266], DS164[267], DS164[269], DS164[270], DS164[272], DS164[275], DS164[280], DS164[284], DS164[287], DS164[288], DS164[289], DS164[290], DS164[292], DS164[295], DS164[300], DS164[317], DS164[330], DS164[331], DS164[336], DS164[341], DS164[342], DS164[343], DS164[346], DS164[347], DS164[348], DS164[350], DS164[351], DS164[352], DS164[354], DS164[356], DS164[357], DS164[359], DS164[360], DS164[361], DS164[364], DS164[365], DS164[367], DS164[368], DS164[370], DS164[371], DS164[379], DS164[381], DS164[392], DS164[395], DS164[396], DS164[401], DS164[404], DS164[405], DS164[410], DS164[411], DS164[413], DS164[414], DS164[419], DS164[421], DS164[422], DS164[423], DS164[424], DS164[425], DS164[427], DS164[430], DS164[432], DS164[434], DS164[435], DS164[437], DS164[438], DS164[439], DS164[440], DS164[441], DS164[442], DS164[443], DS164[445], DS164[446], DS164[447], DS164[448], DS164[449], DS164[451], DS164[452], DS164[453], DS164[455], DS164[457], DS164[458], DS164[460], DS164[461], DS164[464], DS164[470], DS164[471], DS164[472], DS164[475], DS164[476], DS164[477], DS164[479], DS164[482], DS164[483], DS164[484], DS164[485], DS164[486], DS164[490], DS164[496], DS164[497], DS164[498], DS164[501], DS164[502], DS164[508], DS164[509], DS164[510], DS164[511], DS164[514], DS164[517], DS164[519], DS164[520], DS164[531], DS164[532], DS164[533], DS164[534], DS164[537], DS164[538], DS164[540], DS164[542], DS164[546], DS164[547], DS164[548], DS164[551], DS164[557], DS164[560], DS164[562], DS164[566], DS164[567], DS164[568], DS164[569], DS164[571], DS164[572], DS164[573], DS164[574], DS164[575], DS164[577], DS164[578], DS164[579], DS164[580], DS164[581], DS164[582], DS164[583], DS164[584], DS164[585], DS164[586], DS164[589], DS164[590], DS164[592], DS164[593], DS164[594], DS164[596], DS164[598], DS164[599], DS164[600], DS164[603], DS164[604], DS164[605], DS164[606], DS164[607], DS164[608], DS164[612], DS164[613], DS164[615], DS164[616], DS164[624], DS164[627], DS164[628], DS164[633], DS164[634], DS164[635], DS164[636], DS164[639], DS164[640], DS164[641], DS164[642], DS164[643], DS164[644], DS164[646], DS164[647], DS164[648], DS164[650], DS164[651], DS164[653], DS164[654], DS164[655], DS164[656], DS164[661], DS164[664], DS164[666], DS164[668], DS164[673], DS164[676], DS164[682], DS164[683], DS164[685], DS164[690], DS164[695], DS164[696], DS164[697], DS164[698], DS164[701], DS164[703], DS164[704], DS164[709], DS164[710], DS164[711], DS164[714], DS164[715], DS164[716], DS164[718], DS164[719], DS164[720], DS164[721], DS164[722], DS164[724], DS164[725], DS164[727], DS164[728], DS164[729], DS164[730], DS164[731], DS164[734], DS164[735], DS164[736], DS164[737], DS164[738], DS164[739], DS164[741], DS164[742], DS164[746], DS164[748], DS164[749], DS164[750], DS164[751], DS164[752], DS164[753], DS164[754], DS164[755], DS164[759], DS164[760], DS164[762], DS164[763], DS164[764], DS164[765], DS164[766], DS164[767], DS164[768], DS164[769], DS164[770], DS164[774], DS164[775], DS164[777], DS164[778], DS164[779], DS164[780], DS164[781], DS164[782], DS164[783], DS164[784], DS164[785], DS164[789], DS164[792], DS164[793], DS164[796], DS164[799], DS164[801], DS164[802], DS164[803], DS164[804], DS164[807], DS164[808], DS164[809], DS164[810], DS164[813], DS164[815], DS164[816], DS164[817], DS164[820], DS164[822], DS164[823], DS164[824], DS164[825], DS164[826], DS164[827], DS164[828], DS164[833], DS164[835], DS164[836], DS164[837], DS164[838], DS164[845], DS164[848], DS164[849], DS164[851], DS164[855], DS164[857], DS164[862], DS164[864], DS164[866], DS164[870], DS164[871], DS164[872], DS164[876], DS164[879], DS164[880], DS164[881], DS164[882], DS164[883], DS164[884], DS164[885], DS164[886], DS164[887], DS164[888], DS164[889], DS164[890], DS164[891], DS164[892], DS164[893], DS164[894], DS164[895], DS164[896], DS164[897], DS164[898], DS164[899], DS164[900], DS164[901], DS164[902], DS164[903], DS164[904], DS164[905], DS164[906], DS164[907], DS164[908], DS164[909], DS164[910], DS164[913], DS164[922], DS164[925], DS164[926], DS164[927], DS164[932], DS164[937], DS164[941], DS164[946], DS164[947], DS164[948], DS164[952], DS164[953], DS164[955], DS164[956], DS164[957], DS164[958], DS164[959], DS164[960], DS164[961], DS164[962], DS164[963], DS164[964], DS164[967], DS164[971], DS164[973], DS164[976], DS164[977], DS164[978], DS164[981], DS164[982], DS164[983], DS164[984], DS164[985], DS164[987], DS164[988], DS164[990], DS164[993], DS164[994], DS164[996], DS164[997], DS164[999], DS164[1003], DS164[1004], DS164[1005], DS164[1010], DS164[1012], DS164[1013], DS164[1014], DS164[1015], DS164[1016], DS164[1019], DS164[1021], DS164[1022], DS164[1023], DS164[1024], DS164[1026], DS164[1028], DS164[1033], DS164[1034], DS164[1036], DS164[1037], DS164[1038], DS164[1041], DS164[1042], DS164[1045], DS164[1046], DS164[1050], DS164[1051], DS164[1053], DS164[1055], DS164[1056], DS164[1057], DS164[1059], DS164[1061], DS164[1062], DS164[1063], DS164[1064], DS164[1065], DS164[1066], DS164[1067], DS164[1068], DS164[1069], DS164[1070], DS164[1071], DS164[1072], DS164[1073], DS164[1074], DS164[1075], DS164[1076], DS164[1077], DS164[1078], DS164[1079], DS164[1080], DS164[1081], DS164[1082], DS164[1083], DS164[1084], DS164[1086], DS164[1088], DS164[1091], DS164[1097], DS164[1100], DS164[1101], DS164[1102], DS164[1103], DS164[1104], DS164[1108], DS164[1110], DS164[1111], DS164[1113], DS164[1118], DS164[1120], DS164[1122], DS164[1123], DS164[1124], DS164[1125], DS164[1126], DS164[1127], DS164[1128], DS164[1129], DS164[1130], DS164[1131], DS164[1132], DS164[1133], DS164[1134], DS164[1135], DS164[1136], DS164[1137], DS164[1138], DS164[1139], DS164[1140], DS164[1141], DS164[1142], DS164[1143], DS164[1144], DS164[1145], DS164[1146], DS164[1147], DS164[1148], DS164[1149], DS164[1150], DS164[1151], DS164[1152], DS164[1153], DS164[1154], DS164[1155], DS164[1156], DS164[1157], DS164[1158], DS164[1159], DS164[1160], DS164[1161], DS164[1162], DS164[1163], DS164[1164], DS164[1165], DS164[1166], DS164[1167], DS164[1168], DS164[1169], DS164[1170], DS164[1171], DS164[1172], DS164[1173], DS164[1174], DS164[1175], DS164[1179], DS164[1184], DS164[1185], DS164[1188], DS164[1189], DS164[1190], DS164[1191], DS164[1192], DS164[1193], DS164[1194], DS164[1195], DS164[1196], DS164[1197], DS164[1198], DS164[1199], DS164[1200], DS164[1201], DS164[1202], DS164[1203], DS164[1204], DS164[1205], DS164[1206], DS164[1207], DS164[1208], DS164[1209], DS164[1210], DS164[1211], DS164[1212], DS164[1213], DS164[1214], DS164[1215], DS164[1216], DS164[1217], DS164[1218], DS164[1219], DS164[1220], DS164[1221], DS164[1222], DS164[1223], DS164[1224], DS164[1225], DS164[1226], DS164[1227], DS164[1228], DS164[1229], DS164[1230], DS164[1231], DS164[1234], DS164[1235], DS164[1236], DS164[1238], DS164[1240], DS164[1241], DS164[1242], DS164[1243], DS164[1244], DS164[1245], DS164[1246], DS164[1247], DS164[1248], DS164[1249], DS164[1250], DS164[1251], DS164[1252], DS164[1253], DS164[1254], DS164[1255], DS164[1256], DS164[1257], DS164[1258], DS164[1259], DS164[1260], DS164[1261], DS164[1262], DS164[1263], DS164[1264], DS164[1265], DS164[1266], DS164[1267], DS164[1268], DS164[1269], DS164[1270], DS164[1271], DS164[1272], DS164[1273], DS164[1274], DS164[1275], DS164[1276], DS164[1277], DS164[1278], DS164[1279], DS164[1280], DS164[1281], DS164[1282], DS164[1283], DS164[1284], DS164[1285], DS164[1286], DS164[1287], DS164[1288], DS164[1289], DS164[1290], DS164[1291], DS164[1292], DS164[1293], DS164[1294], DS164[1295], DS164[1296], DS164[1297], DS164[1298], DS164[1299], DS164[1300], DS164[1301], DS164[1304], DS164[1305], DS164[1306], DS164[1307], DS164[1312], DS164[1313], DS164[1314], DS164[1315], DS164[1316], DS164[1317], DS164[1318], DS164[1319], DS164[1320], DS164[1321], DS164[1322], DS164[1323], DS164[1324], DS164[1325], DS164[1326], DS164[1327], DS164[1328], DS164[1329], DS164[1330], DS164[1331], DS164[1332], DS164[1333], DS164[1334], DS164[1335], DS164[1336], DS164[1337], DS164[1338], DS164[1339], DS164[1340], DS164[1341], DS164[1342], DS164[1343], DS164[1344], DS164[1345], DS164[1346], DS164[1347], DS164[1348], DS164[1349], DS164[1350], DS164[1351], DS164[1352], DS164[1353], DS164[1354], DS164[1355], DS164[1356], DS164[1357], DS164[1358], DS164[1359], DS164[1360], DS164[1361], DS164[1362], DS164[1363], DS164[1364], DS164[1365], DS164[1366], DS164[1367], DS164[1368], DS164[1369], DS164[1370], DS164[1371], DS164[1372], DS164[1373], DS164[1374], DS164[1375], DS164[1376], DS164[1377], DS164[1378], DS164[1379], DS164[1380], DS164[1381], DS164[1382], DS164[1383], DS164[1384], DS164[1385], DS164[1386], DS164[1387], DS164[1388], DS164[1389], DS164[1390], DS164[1391], DS164[1392], DS164[1393], DS164[1394], DS164[1395], DS164[1396], DS164[1397], DS164[1398], DS164[1399], DS164[1400], DS164[1401], DS164[1402], DS164[1403], DS164[1404], DS164[1405], DS164[1406], DS164[1407], DS164[1408], DS164[1409], DS164[1410], DS164[1411], DS164[1412], DS164[1413], DS164[1419], DS164[1421], DS164[1422], DS164[1426], DS164[1427], DS164[1428], DS164[1429], DS164[1430], DS164[1431], DS164[1432], DS164[1433], DS164[1434], DS164[1435], DS164[1436], DS164[1437], DS164[1438], DS164[1439], DS164[1440], DS164[1441], DS164[1442], DS164[1443], DS164[1444], DS164[1445], DS164[1446], DS164[1447], DS164[1448], DS164[1449], DS164[1450], DS164[1451], DS164[1452], DS164[1453], DS164[1454], DS164[1455], DS164[1456], DS164[1457], DS164[1458], DS164[1459], DS164[1460], DS164[1461], DS164[1462], DS164[1463], DS164[1464], DS164[1465], DS164[1466], DS164[1467], DS164[1468], DS164[1469], DS164[1470], DS164[1471], DS164[1472], DS164[1473], DS164[1474], DS164[1475], DS164[1476], DS164[1477], DS164[1478], DS164[1479], DS164[1480], DS164[1481], DS164[1482], DS164[1483], DS164[1484], DS164[1485], DS164[1486], DS164[1487], DS164[1488], DS164[1489], DS164[1490], DS164[1491], DS164[1492], DS164[1493], DS164[1494], DS164[1495], DS164[1496], DS164[1497], DS164[1498], DS164[1499], DS164[1500], DS164[1501], DS164[1502], DS164[1503], DS164[1504], DS164[1505], DS164[1506], DS164[1507], DS164[1508], DS164[1509], DS164[1510], DS164[1511], DS164[1512], DS164[1513], DS164[1514], DS164[1515], DS164[1516], DS164[1517], DS164[1518], DS164[1519], DS164[1520], DS164[1521], DS164[1522], DS164[1523], DS164[1524], DS164[1525], DS164[1526], DS164[1527], DS164[1528], DS164[1529], DS164[1530], DS164[1531], DS164[1532], DS164[1533], DS164[1534], DS164[1535], DS164[1536], DS164[1537], DS164[1538], DS164[1539], DS164[1540], DS164[1541], DS164[1542], DS164[1543], DS164[1544], DS164[1545], DS164[1546], DS164[1547], DS164[1548], DS164[1549], DS164[1550], DS164[1551], DS164[1552], DS164[1553], DS164[1554], DS164[1555], DS164[1556], DS164[1557], DS164[1558]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,164]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,164] is 184 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,164] is RDS164:=[ [ 4, 5, 6, 8, 9, 13, 15, 17, 18, 21, 22, 23, 25, 28, 33, 36, 38, 39, 44, 45, 47, 50, 55, 56, 60, 61, 62, 63 ], [ 3, 4, 6, 8, 9, 13, 15, 16, 17, 18, 20, 22, 23, 25, 29, 33, 38, 39, 42, 47, 49, 55, 58, 59, 60, 61, 62, 63 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 21, 22, 24, 26, 28, 33, 34, 38, 41, 43, 44, 45, 47, 50, 53, 57, 62, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 21, 22, 23, 24, 26, 28, 33, 34, 38, 41, 47, 50, 53, 57, 58, 59, 60, 62 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 19, 21, 22, 23, 28, 33, 34, 38, 40, 43, 46, 47, 48, 50, 53, 60, 62, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 19, 21, 22, 23, 27, 33, 34, 38, 40, 46, 47, 48, 51, 53, 58, 59, 60, 62 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 19, 21, 22, 23, 25, 28, 33, 34, 38, 40, 43, 47, 50, 53, 60, 61, 62, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 33, 34, 38, 40, 47, 51, 53, 58, 59, 60, 61, 62 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 24, 26, 28, 33, 38, 39, 43, 44, 45, 47, 50, 55, 62, 63, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 28, 33, 35, 38, 39, 43, 46, 47, 48, 50, 54, 60, 62, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 27, 33, 35, 38, 39, 46, 47, 48, 51, 54, 58, 59, 60, 62 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 25, 28, 33, 35, 38, 39, 43, 47, 50, 54, 60, 61, 62, 64 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 25, 27, 33, 35, 38, 39, 47, 51, 54, 58, 59, 60, 61, 62 ], [ 3, 4, 5, 6, 8, 9, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 28, 33, 38, 39, 47, 50, 55, 58, 59, 60, 62, 63 ], [ 3, 4, 5, 6, 8, 9, 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