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Deficiency minus one presentation for the Tits Group

Submitted by Paul Timmons on Sun, 05/02/2021 - 04:27.
Note that the simple group T (named for Jacques Tits) of order 17971200
has a transitive permutation representation on 2304 points.

A general formula for the number of edges in a n-cube is n.2^(n-1) to
be found in this extract from "Beyond the Third Dimension" by Thomas Banchoff
https://www.math.brown.edu/tbanchof/Beyond3d/chapter4/section05.html

It is worth mentioning another source here for a general formula
for the number of pieces of different types on a d-dimensional Rubiks' cube
http://www.gravitation3d.com/magiccube5d/anatomy.html

It can be seen that substituting n=9 gives us the required number of points
(or edges of a nine-dimensional Rubik's cube with 3 divisions per side).

It is natural to ask whether we may find a presentation for T with
particular geometric position (e.g. reflection through the centroid) at
a maximum distance or near maximal distance given the choice of generators
(similar to the thread Presentation for the Mathieu Group M24 from dedge superflip )

I found a presentation for T generated by a and b (see below for explicit values)
which to my knowledge is more efficient then any known presentation to date.
All computations were performed with the Monoid Automata Factory software.

Tits group 2F(4)' - Sporadic simple group (not linear)
As the edges of a nine dimensional Rubik's cube (which expose eight visible stickers)
(Note: 2^8*Binomial(9,8) = 2304 points using GAP)

The simple group T can be presented as three relator quotient of C(2)*C(4), i.e.



< a,b | a2 = b4 = 1,


(ab)13 = 1,


[a,bab]2 = 1,


(abab-1)2(ab)3ab2ab-1ab2abab2(ab-1)3(abab-1)2 = 1 >

or as a deficiency minus one presentation



< a,b | a2b4,
 
(ab)9ab2ab(ab-1)2abab2,
 
(abab-1)2(ab)3ab2ab-1ab2abab2(ab-1)3(abab-1)2 = 1 >

by combining the first relator with the second and the third with the fourth in the first presentation
making a total presentation length of 79.

Antipodes for the Cayley graph of T=<a,b> are wb and its inverse wb^-1 both of order four and length 49.
where



w = ab2ababab2ab2ab-1abab2abab-1abab2ababab-1abab2abab-1ab2a

Note also (wb)^2 = b^2. The value w can be chosen such that it is a reflection
through the centroid (or centre of gravity) so that all edges get moved to the unique
furthest away edge on the 9D cube.

Finally the values of a,b and w as permutations in GAP:

a:=
(   1, 659)(   2,2249)(   3,2146)(   4,1247)(   5, 710)(   6,1080)(   7, 855)(   8,1687)
(   9, 817)(  10,2276)(  11,2252)(  12, 658)(  13,2145)(  14,1246)(  15,1942)(  16,1884)
(  17,2108)(  18, 645)(  19,2071)(  20, 118)(  21,1696)(  22,1309)(  23, 271)(  24, 720)
(  25, 636)(  26,  96)(  27,1350)(  28,1532)(  29, 294)(  30, 216)(  31, 963)(  32, 351)
(  33, 459)(  34,1164)(  35, 516)(  36,1756)(  37,1145)(  38,2095)(  39, 793)(  40,2097)
(  41, 675)(  42,1446)(  43, 356)(  44,1488)(  45, 842)(  46, 939)(  47, 555)(  48, 101)
(  49, 212)(  50, 766)(  51,1818)(  52,2043)(  53,1141)(  54,2128)(  55, 603)(  56,1152)
(  57,1038)(  58, 773)(  59, 684)(  60, 535)(  61,1637)(  62, 543)(  63,1969)(  64,1289)
(  65,1896)(  66,1251)(  67, 648)(  68,  90)(  70,1566)(  71,2040)(  72,1788)(  73,  74)
(  75,1998)(  76, 327)(  77,2202)(  78, 260)(  79,1508)(  80, 298)(  82,2046)(  83,2217)
(  84,1995)(  85,1323)(  86,1919)(  87, 240)(  88,1384)(  89,1590)(  91, 372)(  92, 913)
(  93,1948)(  94, 431)(  95, 799)(  97,1682)(  98,1181)(  99,1906)( 100,1779)( 102,1541)
( 103,2302)( 104, 190)( 105, 859)( 106,1850)( 107,1517)( 108,1380)( 109, 718)( 110,1801)
( 111,1150)( 112,2303)( 113,1234)( 114,1051)( 115,1915)( 116, 832)( 117, 982)( 119,1326)
( 120, 461)( 121, 441)( 122,1201)( 123, 662)( 124, 727)( 125, 979)( 127,1110)( 128,2177)
( 129,1001)( 130,1885)( 131,1632)( 132,1114)( 133,1825)( 134, 666)( 135,2295)( 136,1435)
( 137,1618)( 138, 624)( 139,2298)( 140,1512)( 141,1932)( 142, 442)( 143,1819)( 144, 925)
( 145,2006)( 146, 777)( 147,1857)( 148, 491)( 149,1426)( 150,1305)( 151,1314)( 152, 248)
( 153, 717)( 154,1358)( 155,1136)( 156,1524)( 157,2274)( 158,2271)( 159, 296)( 160, 814)
( 161,2015)( 162, 539)( 163,1680)( 164,1662)( 166,2086)( 167, 500)( 168, 205)( 169,1291)
( 170,1338)( 171,1147)( 172, 462)( 173,1760)( 174,1056)( 175, 848)( 177,2204)( 178,1050)
( 179,2199)( 180,1961)( 181,1642)( 182,2248)( 183,1342)( 184,1442)( 185,1008)( 186, 949)
( 187, 692)( 188,1418)( 189,1016)( 191,2290)( 192, 405)( 193, 378)( 195,2257)( 196, 456)
( 197,1874)( 198,2261)( 199,1593)( 200, 644)( 201,2120)( 202,1028)( 203,1250)( 204, 802)
( 206, 724)( 207,1215)( 208, 865)( 209,1594)( 210,1860)( 213, 797)( 214,1805)( 215,1241)
( 217, 860)( 218, 869)( 219,2266)( 220,1034)( 221,1572)( 222, 725)( 223, 517)( 224,1020)
( 226,2300)( 227,1683)( 228,2024)( 230,2167)( 231,1176)( 232,1129)( 233, 745)( 234,1226)
( 235, 770)( 236, 960)( 237,2051)( 238,2107)( 239,1897)( 241, 686)( 242,2019)( 243,1074)
( 244,1231)( 245,1848)( 246,1290)( 247,1166)( 249,1236)( 250,1144)( 252,1261)( 253,1748)
( 254, 873)( 255,1559)( 256, 596)( 257, 434)( 258,1064)( 259,1415)( 261,1900)( 262, 504)
( 263, 358)( 264,1336)( 265,2114)( 266, 433)( 267,1322)( 268,1441)( 269,1526)( 270,1334)
( 272, 674)( 273,1894)( 274,1355)( 275,1131)( 276,1172)( 277,1473)( 278, 279)( 280,1284)
( 281,1095)( 282, 407)( 283,1088)( 284,1750)( 285,1252)( 286,1898)( 287, 739)( 288,2123)
( 289,1603)( 290,1146)( 292, 300)( 293, 988)( 295,1424)( 297,1723)( 299,1327)( 301, 511)
( 302,2075)( 303, 681)( 304,1772)( 305, 664)( 306, 880)( 307, 677)( 309, 803)( 310,2117)
( 311,1429)( 312,1300)( 313,2200)( 314, 554)( 315,1458)( 316, 736)( 317,1549)( 318,1892)
( 319,1127)( 320,1755)( 321,1219)( 322, 503)( 323, 973)( 324,1983)( 325,1093)( 326,1427)
( 328,1422)( 329,2098)( 330, 815)( 331, 373)( 332, 446)( 333,2226)( 334,1815)( 335,1952)
( 336,1156)( 337, 508)( 338,1117)( 339, 452)( 340, 353)( 341, 959)( 343,1699)( 344,2091)
( 345,1980)( 346, 687)( 347,1619)( 348, 621)( 349,1545)( 350, 643)( 354,1716)( 357,1537)
( 359,1604)( 360, 629)( 361, 394)( 362,1689)( 363,1165)( 364, 857)( 365,1650)( 366, 728)
( 367,1507)( 368,1588)( 369,1797)( 370,1987)( 371,2052)( 374,1243)( 375, 689)( 376, 755)
( 377, 477)( 379, 445)( 380,1328)( 381,1304)( 382, 875)( 383, 758)( 384,1167)( 385,2242)
( 386, 512)( 387, 740)( 388, 630)( 389,1864)( 390,1638)( 391, 741)( 392,2034)( 393,1393)
( 395,1560)( 396, 564)( 397,1846)( 398, 542)( 399,1168)( 400,1739)( 401, 451)( 403,1711)
( 404, 838)( 406, 944)( 408,1408)( 409,2153)( 410,1474)( 411,2090)( 412, 544)( 413,2082)
( 414, 957)( 415, 518)( 416,1266)( 417, 584)( 418,1921)( 419,1520)( 420,1778)( 421,1954)
( 422,1580)( 423, 425)( 424, 792)( 426,1402)( 427,2012)( 428,2136)( 429,2250)( 430, 470)
( 432,1084)( 435,1707)( 436,1257)( 437,2063)( 438,1090)( 439, 598)( 440,1079)( 443,1909)
( 444,2081)( 447,1367)( 448,2245)( 449,1727)( 450, 978)( 453, 484)( 454,2045)( 455, 889)
( 457, 909)( 458,1544)( 460,1274)( 463,1040)( 464,1450)( 465,1449)( 466,1498)( 467,1205)
( 468,1777)( 469, 656)( 471,1789)( 472,1210)( 473,1399)( 474,1297)( 475,1230)( 476,1348)
( 478,1372)( 479,1664)( 480,1159)( 481,1343)( 482,1773)( 483,1496)( 485,1865)( 486,2182)
( 487, 507)( 488, 668)( 490,1854)( 492, 498)( 493,1908)( 494, 884)( 495,2247)( 496,1185)
( 497,1599)( 499, 788)( 501, 946)( 502,1026)( 505,1456)( 506,1963)( 509, 877)( 510,1871)
( 513,1718)( 514,2054)( 515,1014)( 519,1263)( 520,1416)( 521, 926)( 522,1690)( 523,1970)
( 524,2281)( 525,2164)( 526,2092)( 527, 743)( 528,1170)( 529, 663)( 530,1480)( 531,1153)
( 532,1631)( 533,1271)( 534,1922)( 536, 772)( 537,2233)( 538,2103)( 541,2009)( 545,1957)
( 546,1693)( 547,1347)( 549, 846)( 550,1105)( 551, 868)( 552, 908)( 553,2011)( 556,1022)
( 557,2058)( 558,1368)( 559,1242)( 560,1500)( 561,2112)( 562, 594)( 563,1851)( 565, 606)
( 566,1695)( 567,1006)( 568,1182)( 569, 742)( 570,1749)( 571,1100)( 572, 769)( 573, 952)
( 574,1562)( 575,1502)( 576,1775)( 577,1605)( 578,1248)( 579,1332)( 580,2126)( 581,1663)
( 582, 595)( 583, 650)( 585, 735)( 586,1914)( 588,1686)( 589, 693)( 590,1962)( 591, 876)
( 592, 759)( 593,1518)( 597,1653)( 599,2270)( 600,1475)( 601, 924)( 602,2285)( 604, 862)
( 605, 708)( 607,2084)( 608,2212)( 609,1613)( 610,2301)( 611,1578)( 612,1833)( 613, 685)
( 614,1692)( 615,1640)( 616,2283)( 617,1082)( 618,2216)( 619,1648)( 620,2028)( 622, 800)
( 623,2039)( 625,1106)( 626,1490)( 627,1546)( 628, 831)( 631,1471)( 632,1398)( 633,1013)
( 634,1459)( 635,2033)( 637,1434)( 638,2070)( 639,1370)( 640, 661)( 641,2111)( 642,1595)
( 646,2299)( 647,2277)( 649,1039)( 651,1924)( 652,1985)( 653,1655)( 654,2255)( 655,1767)
( 657,1479)( 660,1649)( 665,1971)( 667,1162)( 670,1487)( 671,1902)( 672, 866)( 673,1878)
( 676, 804)( 678,1113)( 679,1943)( 682,1880)( 683,1849)( 688, 723)( 690,1730)( 691,1890)
( 694,1140)( 695,1053)( 696,2010)( 697,1126)( 698, 942)( 699,1228)( 700, 783)( 701,1148)
( 702,1139)( 703,1381)( 704,1339)( 705,1222)( 706,2014)( 707,1770)( 709,2273)( 711, 891)
( 712,1293)( 713,1940)( 714,1558)( 715,1158)( 716,1602)( 719,1516)( 721,1428)( 722,1999)
( 726, 894)( 729,1679)( 730,2288)( 732, 995)( 733,1785)( 734,1783)( 737,1112)( 738,1867)
( 744, 886)( 746, 895)( 747,2209)( 748, 781)( 749,1227)( 750,1330)( 751,1438)( 752,1552)
( 753,1889)( 754,1751)( 756,2260)( 757,1400)( 760,2094)( 761,2185)( 762, 858)( 763,1360)
( 764,1298)( 765,1853)( 767,1048)( 768,1477)( 771,1204)( 774,1852)( 775,1528)( 776,1030)
( 778,1004)( 779,1278)( 780,1548)( 782,1003)( 784,1392)( 785,1268)( 786, 806)( 787, 943)
( 789, 965)( 790,1451)( 791, 843)( 794, 985)( 795,1295)( 796,1795)( 798,1821)( 801,1746)
( 805,1950)( 807, 975)( 808, 933)( 809,2223)( 810,2042)( 811,2180)( 813,1684)( 816,1629)
( 818,2064)( 819,1220)( 820,2073)( 821,1774)( 822,2141)( 823,1996)( 824,1344)( 825,1282)
( 826, 929)( 827,2061)( 828,1437)( 829,1844)( 830,1378)( 833,1277)( 834, 935)( 835,1177)
( 836,1585)( 837,1486)( 839,1639)( 840,1272)( 841,2272)( 844,2279)( 845,2154)( 847,1017)
( 849,1808)( 850,2170)( 851,2008)( 852,1565)( 853,1457)( 856,1461)( 861,2049)( 863,1899)
( 864,2076)( 867,1965)( 870,1455)( 871,1573)( 872,1519)( 874,1359)( 878,1233)( 879,1254)
( 881,1352)( 882,2158)( 883,1245)( 885,2089)( 887,1200)( 888,1705)( 890,1591)( 892,1729)
( 893,2294)( 896,1199)( 897,1806)( 898,2246)( 899,2203)( 900,1354)( 902,2047)( 903,1504)
( 905,1265)( 906,2214)( 907,2269)( 910,1835)( 911,1694)( 912,2229)( 914,2220)( 915,1249)
( 916,1149)( 917,1059)( 918,1281)( 919,1840)( 920,1790)( 921,1611)( 922,2069)( 923,1043)
( 927,1688)( 928,1410)( 930,1505)( 931,2072)( 932,1296)( 934,2208)( 936,1411)( 937,1269)
( 938,1966)( 940, 981)( 941,1945)( 945,2191)( 947,1724)( 948,1303)( 951,1497)( 953,1121)
( 954,1279)( 955, 964)( 958,1403)( 961,1651)( 962,1843)( 966,1916)( 967,1765)( 968,2083)
( 969,1301)( 970,2201)( 971,1255)( 972,1859)( 974,1627)( 976,1669)( 977,2118)( 980,2139)
( 983,1828)( 984,2133)( 986,1325)( 987,1967)( 989,1143)( 990,1374)( 991,1078)( 992,1646)
( 993,1703)( 994,2131)( 997,1647)( 998,1571)( 999,1842)(1000,1873)(1002,2080)(1005,2022)
(1007,1621)(1009,1294)(1010,1678)(1011,2194)(1012,1244)(1015,1275)(1018,2029)(1019,1510)
(1021,1157)(1023,1836)(1024,1223)(1025,1645)(1027,1216)(1029,1769)(1031,1893)(1032,1049)
(1033,2264)(1035,1910)(1036,2215)(1037,2211)(1041,1160)(1042,1592)(1044,1661)(1045,1981)
(1046,2127)(1047,1253)(1052,2137)(1054,1601)(1055,2124)(1057,1673)(1058,1476)(1060,1814)
(1061,1866)(1062,1383)(1065,1771)(1066,1907)(1067,1195)(1068,1203)(1069,1535)(1070,1927)
(1071,2125)(1072,1644)(1073,1316)(1075,1076)(1077,1895)(1081,1597)(1083,1587)(1085,1191)
(1086,1364)(1087,1731)(1089,1764)(1091,2263)(1092,1193)(1094,2085)(1096,1419)(1097,1568)
(1098,1443)(1099,2074)(1101,2278)(1102,1482)(1103,1208)(1104,1780)(1107,1872)(1108,2152)
(1109,1320)(1111,1262)(1115,2166)(1116,1812)(1118,1453)(1119,1491)(1120,1691)(1122,1855)
(1123,1288)(1124,2256)(1125,1586)(1128,1192)(1130,2079)(1132,1493)(1133,2192)(1134,2025)
(1135,1273)(1137,1574)(1138,1986)(1142,1531)(1151,2130)(1154,1285)(1155,1439)(1161,1810)
(1163,1614)(1169,1732)(1171,2210)(1173,1633)(1175,1719)(1178,2065)(1180,1722)(1183,1989)
(1186,2026)(1187,1811)(1188,1870)(1189,1276)(1190,1579)(1194,1616)(1196,1329)(1197,1551)
(1202,2102)(1206,2265)(1207,1615)(1209,2234)(1211,1654)(1212,1376)(1213,1994)(1214,1492)
(1217,1736)(1218,1530)(1221,2157)(1224,1761)(1225,2007)(1229,1447)(1232,1466)(1235,1554)
(1238,1757)(1239,1382)(1256,2110)(1258,1321)(1259,2240)(1260,1674)(1264,1982)(1267,1478)
(1270,1371)(1283,1706)(1286,1534)(1287,2172)(1292,2161)(1299,2280)(1306,2129)(1307,1635)
(1308,2205)(1310,1318)(1311,1939)(1312,1912)(1313,2188)(1315,1514)(1317,1817)(1319,1385)
(1324,2162)(1331,1887)(1333,1575)(1335,1538)(1337,1397)(1341,2189)(1345,1829)(1346,1715)
(1349,1926)(1351,1841)(1353,2115)(1356,2144)(1357,1953)(1361,2066)(1362,2000)(1363,1463)
(1365,2235)(1366,1991)(1373,1659)(1377,1596)(1379,2032)(1386,2121)(1387,1569)(1388,1469)
(1389,1698)(1390,1656)(1391,2160)(1394,2018)(1395,1612)(1396,1665)(1401,1577)(1404,2178)
(1405,1834)(1406,2296)(1407,1436)(1409,1701)(1412,1792)(1413,1976)(1414,2176)(1417,1883)
(1420,1826)(1421,1977)(1423,1676)(1425,1721)(1430,1540)(1431,1626)(1432,2213)(1433,2243)
(1440,2225)(1444,2151)(1445,1523)(1448,2232)(1454,1666)(1460,1550)(1462,2135)(1464,2259)
(1465,1803)(1467,2231)(1468,1576)(1472,1960)(1481,1543)(1483,1561)(1484,2050)(1485,1710)
(1494,2067)(1495,1791)(1499,1968)(1501,1920)(1503,2035)(1506,1625)(1509,2186)(1511,2027)
(1513,1802)(1515,1725)(1521,2289)(1522,1667)(1525,1800)(1527,2241)(1529,1553)(1533,2254)
(1536,1928)(1539,1610)(1542,1589)(1547,1685)(1555,1570)(1557,1929)(1563,1752)(1564,2275)
(1567,1583)(1581,1813)(1582,2088)(1584,1776)(1598,1623)(1600,1620)(1606,2013)(1607,2198)
(1608,2060)(1609,1726)(1622,1816)(1624,1737)(1628,1931)(1630,2236)(1634,2238)(1636,1936)
(1641,1657)(1643,1733)(1652,1768)(1658,1925)(1660,2143)(1668,2244)(1670,1975)(1671,2197)
(1672,2291)(1675,1838)(1677,1681)(1697,1947)(1700,2053)(1702,2104)(1704,2258)(1708,1972)
(1709,2183)(1712,1758)(1713,2267)(1714,1820)(1717,2163)(1720,2048)(1728,2222)(1734,2228)
(1738,1941)(1740,1911)(1741,2150)(1742,2099)(1743,2078)(1744,2147)(1745,1804)(1747,2181)
(1753,2239)(1754,1951)(1759,2219)(1762,1876)(1763,2062)(1766,2196)(1781,2171)(1782,2005)
(1784,2195)(1786,2287)(1787,2003)(1793,1997)(1794,2132)(1796,1837)(1798,2184)(1799,1923)
(1807,2148)(1809,2055)(1822,1863)(1823,2156)(1824,1839)(1827,1845)(1830,2221)(1831,1847)
(1832,2284)(1858,1944)(1861,2218)(1862,2116)(1868,1904)(1869,1964)(1875,1930)(1877,1956)
(1879,1988)(1882,2056)(1886,2077)(1888,2253)(1891,1946)(1901,2119)(1903,2268)(1905,2031)
(1918,2190)(1933,2206)(1934,1974)(1935,2286)(1937,2093)(1938,2224)(1949,2251)(1955,2138)
(1959,2293)(1973,2017)(1978,2021)(1979,2105)(1984,2262)(1990,2193)(1992,2227)(1993,2292)
(2002,2168)(2016,2174)(2020,2187)(2036,2037)(2041,2304)(2044,2101)(2057,2149)(2059,2142)
(2100,2237)(2106,2230)(2109,2173)(2113,2155)(2122,2140)(2134,2282)(2159,2169)(2165,2297);


b:=
(   1,   2)(   3,   4)(   5,   6)(   7,   8)(   9,  10)(  11,  12)(  13,  14)(  15,  16)
(  17,  18)(  19,  20)(  21,  22)(  23,  24)(  25,  26)(  27,  28)(  29,  30)(  31,  32)
(  33,  34)(  35,  36)(  37,  38)(  39,  40)(  41,  42)(  43,  44)(  45,  46)(  47,  48)
(  49,  50)(  51,  52)(  53,  54)(  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,  83,  84)(  85,  86,  87,  88)(  89,  90,  91,  92)(  93,  94,  95,  96)
(  97,  98,  99, 100)( 101, 102, 103, 104)( 105, 106, 107, 108)( 109, 110, 111, 112)
( 113, 114, 115, 116)( 117, 118, 119, 120)( 121, 122, 123, 124)( 125, 126, 127, 128)
( 129, 130, 131, 132)( 133, 134, 135, 136)( 137, 138, 139, 140)( 141, 142, 143, 144)
( 145, 146, 147, 148)( 149, 150, 151, 152)( 153, 154, 155, 156)( 157, 158, 159, 160)
( 161, 162, 163, 164)( 165, 166, 167, 168)( 169, 170, 171, 172)( 173, 174, 175, 176)
( 177, 178, 179, 180)( 181, 182, 183, 184)( 185, 186, 187, 188)( 189, 190, 191, 192)
( 193, 194, 195, 196)( 197, 198, 199, 200)( 201, 202, 203, 204)( 205, 206, 207, 208)
( 209, 210, 211, 212)( 213, 214, 215, 216)( 217, 218, 219, 220)( 221, 222, 223, 224)
( 225, 226, 227, 228)( 229, 230, 231, 232)( 233, 234, 235, 236)( 237, 238, 239, 240)
( 241, 242, 243, 244)( 245, 246, 247, 248)( 249, 250, 251, 252)( 253, 254, 255, 256)
( 257, 258, 259, 260)( 261, 262, 263, 264)( 265, 266, 267, 268)( 269, 270, 271, 272)
( 273, 274, 275, 276)( 277, 278, 279, 280)( 281, 282, 283, 284)( 285, 286, 287, 288)
( 289, 290, 291, 292)( 293, 294, 295, 296)( 297, 298, 299, 300)( 301, 302, 303, 304)
( 305, 306, 307, 308)( 309, 310, 311, 312)( 313, 314, 315, 316)( 317, 318, 319, 320)
( 321, 322, 323, 324)( 325, 326, 327, 328)( 329, 330, 331, 332)( 333, 334, 335, 336)
( 337, 338, 339, 340)( 341, 342, 343, 344)( 345, 346, 347, 348)( 349, 350, 351, 352)
( 353, 354, 355, 356)( 357, 358, 359, 360)( 361, 362, 363, 364)( 365, 366, 367, 368)
( 369, 370, 371, 372)( 373, 374, 375, 376)( 377, 378, 379, 380)( 381, 382, 383, 384)
( 385, 386, 387, 388)( 389, 390, 391, 392)( 393, 394, 395, 396)( 397, 398, 399, 400)
( 401, 402, 403, 404)( 405, 406, 407, 408)( 409, 410, 411, 412)( 413, 414, 415, 416)
( 417, 418, 419, 420)( 421, 422, 423, 424)( 425, 426, 427, 428)( 429, 430, 431, 432)
( 433, 434, 435, 436)( 437, 438, 439, 440)( 441, 442, 443, 444)( 445, 446, 447, 448)
( 449, 450, 451, 452)( 453, 454, 455, 456)( 457, 458, 459, 460)( 461, 462, 463, 464)
( 465, 466, 467, 468)( 469, 470, 471, 472)( 473, 474, 475, 476)( 477, 478, 479, 480)
( 481, 482, 483, 484)( 485, 486, 487, 488)( 489, 490, 491, 492)( 493, 494, 495, 496)
( 497, 498, 499, 500)( 501, 502, 503, 504)( 505, 506, 507, 508)( 509, 510, 511, 512)
( 513, 514, 515, 516)( 517, 518, 519, 520)( 521, 522, 523, 524)( 525, 526, 527, 528)
( 529, 530, 531, 532)( 533, 534, 535, 536)( 537, 538, 539, 540)( 541, 542, 543, 544)
( 545, 546, 547, 548)( 549, 550, 551, 552)( 553, 554, 555, 556)( 557, 558, 559, 560)
( 561, 562, 563, 564)( 565, 566, 567, 568)( 569, 570, 571, 572)( 573, 574, 575, 576)
( 577, 578, 579, 580)( 581, 582, 583, 584)( 585, 586, 587, 588)( 589, 590, 591, 592)
( 593, 594, 595, 596)( 597, 598, 599, 600)( 601, 602, 603, 604)( 605, 606, 607, 608)
( 609, 610, 611, 612)( 613, 614, 615, 616)( 617, 618, 619, 620)( 621, 622, 623, 624)
( 625, 626, 627, 628)( 629, 630, 631, 632)( 633, 634, 635, 636)( 637, 638, 639, 640)
( 641, 642, 643, 644)( 645, 646, 647, 648)( 649, 650, 651, 652)( 653, 654, 655, 656)
( 657, 658, 659, 660)( 661, 662, 663, 664)( 665, 666, 667, 668)( 669, 670, 671, 672)
( 673, 674, 675, 676)( 677, 678, 679, 680)( 681, 682, 683, 684)( 685, 686, 687, 688)
( 689, 690, 691, 692)( 693, 694, 695, 696)( 697, 698, 699, 700)( 701, 702, 703, 704)
( 705, 706, 707, 708)( 709, 710, 711, 712)( 713, 714, 715, 716)( 717, 718, 719, 720)
( 721, 722, 723, 724)( 725, 726, 727, 728)( 729, 730, 731, 732)( 733, 734, 735, 736)
( 737, 738, 739, 740)( 741, 742, 743, 744)( 745, 746, 747, 748)( 749, 750, 751, 752)
( 753, 754, 755, 756)( 757, 758, 759, 760)( 761, 762, 763, 764)( 765, 766, 767, 768)
( 769, 770, 771, 772)( 773, 774, 775, 776)( 777, 778, 779, 780)( 781, 782, 783, 784)
( 785, 786, 787, 788)( 789, 790, 791, 792)( 793, 794, 795, 796)( 797, 798, 799, 800)
( 801, 802, 803, 804)( 805, 806, 807, 808)( 809, 810, 811, 812)( 813, 814, 815, 816)
( 817, 818, 819, 820)( 821, 822, 823, 824)( 825, 826, 827, 828)( 829, 830, 831, 832)
( 833, 834, 835, 836)( 837, 838, 839, 840)( 841, 842, 843, 844)( 845, 846, 847, 848)
( 849, 850, 851, 852)( 853, 854, 855, 856)( 857, 858, 859, 860)( 861, 862, 863, 864)
( 865, 866, 867, 868)( 869, 870, 871, 872)( 873, 874, 875, 876)( 877, 878, 879, 880)
( 881, 882, 883, 884)( 885, 886, 887, 888)( 889, 890, 891, 892)( 893, 894, 895, 896)
( 897, 898, 899, 900)( 901, 902, 903, 904)( 905, 906, 907, 908)( 909, 910, 911, 912)
( 913, 914, 915, 916)( 917, 918, 919, 920)( 921, 922, 923, 924)( 925, 926, 927, 928)
( 929, 930, 931, 932)( 933, 934, 935, 936)( 937, 938, 939, 940)( 941, 942, 943, 944)
( 945, 946, 947, 948)( 949, 950, 951, 952)( 953, 954, 955, 956)( 957, 958, 959, 960)
( 961, 962, 963, 964)( 965, 966, 967, 968)( 969, 970, 971, 972)( 973, 974, 975, 976)
( 977, 978, 979, 980)( 981, 982, 983, 984)( 985, 986, 987, 988)( 989, 990, 991, 992)
( 993, 994, 995, 996)( 997, 998, 999,1000)(1001,1002,1003,1004)(1005,1006,1007,1008)
(1009,1010,1011,1012)(1013,1014,1015,1016)(1017,1018,1019,1020)(1021,1022,1023,1024)
(1025,1026,1027,1028)(1029,1030,1031,1032)(1033,1034,1035,1036)(1037,1038,1039,1040)
(1041,1042,1043,1044)(1045,1046,1047,1048)(1049,1050,1051,1052)(1053,1054,1055,1056)
(1057,1058,1059,1060)(1061,1062,1063,1064)(1065,1066,1067,1068)(1069,1070,1071,1072)
(1073,1074,1075,1076)(1077,1078,1079,1080)(1081,1082,1083,1084)(1085,1086,1087,1088)
(1089,1090,1091,1092)(1093,1094,1095,1096)(1097,1098,1099,1100)(1101,1102,1103,1104)
(1105,1106,1107,1108)(1109,1110,1111,1112)(1113,1114,1115,1116)(1117,1118,1119,1120)
(1121,1122,1123,1124)(1125,1126,1127,1128)(1129,1130,1131,1132)(1133,1134,1135,1136)
(1137,1138,1139,1140)(1141,1142,1143,1144)(1145,1146,1147,1148)(1149,1150,1151,1152)
(1153,1154,1155,1156)(1157,1158,1159,1160)(1161,1162,1163,1164)(1165,1166,1167,1168)
(1169,1170,1171,1172)(1173,1174,1175,1176)(1177,1178,1179,1180)(1181,1182,1183,1184)
(1185,1186,1187,1188)(1189,1190,1191,1192)(1193,1194,1195,1196)(1197,1198,1199,1200)
(1201,1202,1203,1204)(1205,1206,1207,1208)(1209,1210,1211,1212)(1213,1214,1215,1216)
(1217,1218,1219,1220)(1221,1222,1223,1224)(1225,1226,1227,1228)(1229,1230,1231,1232)
(1233,1234,1235,1236)(1237,1238,1239,1240)(1241,1242,1243,1244)(1245,1246,1247,1248)
(1249,1250,1251,1252)(1253,1254,1255,1256)(1257,1258,1259,1260)(1261,1262,1263,1264)
(1265,1266,1267,1268)(1269,1270,1271,1272)(1273,1274,1275,1276)(1277,1278,1279,1280)
(1281,1282,1283,1284)(1285,1286,1287,1288)(1289,1290,1291,1292)(1293,1294,1295,1296)
(1297,1298,1299,1300)(1301,1302,1303,1304)(1305,1306,1307,1308)(1309,1310,1311,1312)
(1313,1314,1315,1316)(1317,1318,1319,1320)(1321,1322,1323,1324)(1325,1326,1327,1328)
(1329,1330,1331,1332)(1333,1334,1335,1336)(1337,1338,1339,1340)(1341,1342,1343,1344)
(1345,1346,1347,1348)(1349,1350,1351,1352)(1353,1354,1355,1356)(1357,1358,1359,1360)
(1361,1362,1363,1364)(1365,1366,1367,1368)(1369,1370,1371,1372)(1373,1374,1375,1376)
(1377,1378,1379,1380)(1381,1382,1383,1384)(1385,1386,1387,1388)(1389,1390,1391,1392)
(1393,1394,1395,1396)(1397,1398,1399,1400)(1401,1402,1403,1404)(1405,1406,1407,1408)
(1409,1410,1411,1412)(1413,1414,1415,1416)(1417,1418,1419,1420)(1421,1422,1423,1424)
(1425,1426,1427,1428)(1429,1430,1431,1432)(1433,1434,1435,1436)(1437,1438,1439,1440)
(1441,1442,1443,1444)(1445,1446,1447,1448)(1449,1450,1451,1452)(1453,1454,1455,1456)
(1457,1458,1459,1460)(1461,1462,1463,1464)(1465,1466,1467,1468)(1469,1470,1471,1472)
(1473,1474,1475,1476)(1477,1478,1479,1480)(1481,1482,1483,1484)(1485,1486,1487,1488)
(1489,1490,1491,1492)(1493,1494,1495,1496)(1497,1498,1499,1500)(1501,1502,1503,1504)
(1505,1506,1507,1508)(1509,1510,1511,1512)(1513,1514,1515,1516)(1517,1518,1519,1520)
(1521,1522,1523,1524)(1525,1526,1527,1528)(1529,1530,1531,1532)(1533,1534,1535,1536)
(1537,1538,1539,1540)(1541,1542,1543,1544)(1545,1546,1547,1548)(1549,1550,1551,1552)
(1553,1554,1555,1556)(1557,1558,1559,1560)(1561,1562,1563,1564)(1565,1566,1567,1568)
(1569,1570,1571,1572)(1573,1574,1575,1576)(1577,1578,1579,1580)(1581,1582,1583,1584)
(1585,1586,1587,1588)(1589,1590,1591,1592)(1593,1594,1595,1596)(1597,1598,1599,1600)
(1601,1602,1603,1604)(1605,1606,1607,1608)(1609,1610,1611,1612)(1613,1614,1615,1616)
(1617,1618,1619,1620)(1621,1622,1623,1624)(1625,1626,1627,1628)(1629,1630,1631,1632)
(1633,1634,1635,1636)(1637,1638,1639,1640)(1641,1642,1643,1644)(1645,1646,1647,1648)
(1649,1650,1651,1652)(1653,1654,1655,1656)(1657,1658,1659,1660)(1661,1662,1663,1664)
(1665,1666,1667,1668)(1669,1670,1671,1672)(1673,1674,1675,1676)(1677,1678,1679,1680)
(1681,1682,1683,1684)(1685,1686,1687,1688)(1689,1690,1691,1692)(1693,1694,1695,1696)
(1697,1698,1699,1700)(1701,1702,1703,1704)(1705,1706,1707,1708)(1709,1710,1711,1712)
(1713,1714,1715,1716)(1717,1718,1719,1720)(1721,1722,1723,1724)(1725,1726,1727,1728)
(1729,1730,1731,1732)(1733,1734,1735,1736)(1737,1738,1739,1740)(1741,1742,1743,1744)
(1745,1746,1747,1748)(1749,1750,1751,1752)(1753,1754,1755,1756)(1757,1758,1759,1760)
(1761,1762,1763,1764)(1765,1766,1767,1768)(1769,1770,1771,1772)(1773,1774,1775,1776)
(1777,1778,1779,1780)(1781,1782,1783,1784)(1785,1786,1787,1788)(1789,1790,1791,1792)
(1793,1794,1795,1796)(1797,1798,1799,1800)(1801,1802,1803,1804)(1805,1806,1807,1808)
(1809,1810,1811,1812)(1813,1814,1815,1816)(1817,1818,1819,1820)(1821,1822,1823,1824)
(1825,1826,1827,1828)(1829,1830,1831,1832)(1833,1834,1835,1836)(1837,1838,1839,1840)
(1841,1842,1843,1844)(1845,1846,1847,1848)(1849,1850,1851,1852)(1853,1854,1855,1856)
(1857,1858,1859,1860)(1861,1862,1863,1864)(1865,1866,1867,1868)(1869,1870,1871,1872)
(1873,1874,1875,1876)(1877,1878,1879,1880)(1881,1882,1883,1884)(1885,1886,1887,1888)
(1889,1890,1891,1892)(1893,1894,1895,1896)(1897,1898,1899,1900)(1901,1902,1903,1904)
(1905,1906,1907,1908)(1909,1910,1911,1912)(1913,1914,1915,1916)(1917,1918,1919,1920)
(1921,1922,1923,1924)(1925,1926,1927,1928)(1929,1930,1931,1932)(1933,1934,1935,1936)
(1937,1938,1939,1940)(1941,1942,1943,1944)(1945,1946,1947,1948)(1949,1950,1951,1952)
(1953,1954,1955,1956)(1957,1958,1959,1960)(1961,1962,1963,1964)(1965,1966,1967,1968)
(1969,1970,1971,1972)(1973,1974,1975,1976)(1977,1978,1979,1980)(1981,1982,1983,1984)
(1985,1986,1987,1988)(1989,1990,1991,1992)(1993,1994,1995,1996)(1997,1998,1999,2000)
(2001,2002,2003,2004)(2005,2006,2007,2008)(2009,2010,2011,2012)(2013,2014,2015,2016)
(2017,2018,2019,2020)(2021,2022,2023,2024)(2025,2026,2027,2028)(2029,2030,2031,2032)
(2033,2034,2035,2036)(2037,2038,2039,2040)(2041,2042,2043,2044)(2045,2046,2047,2048)
(2049,2050,2051,2052)(2053,2054,2055,2056)(2057,2058,2059,2060)(2061,2062,2063,2064)
(2065,2066,2067,2068)(2069,2070,2071,2072)(2073,2074,2075,2076)(2077,2078,2079,2080)
(2081,2082,2083,2084)(2085,2086,2087,2088)(2089,2090,2091,2092)(2093,2094,2095,2096)
(2097,2098,2099,2100)(2101,2102,2103,2104)(2105,2106,2107,2108)(2109,2110,2111,2112)
(2113,2114,2115,2116)(2117,2118,2119,2120)(2121,2122,2123,2124)(2125,2126,2127,2128)
(2129,2130,2131,2132)(2133,2134,2135,2136)(2137,2138,2139,2140)(2141,2142,2143,2144)
(2145,2146,2147,2148)(2149,2150,2151,2152)(2153,2154,2155,2156)(2157,2158,2159,2160)
(2161,2162,2163,2164)(2165,2166,2167,2168)(2169,2170,2171,2172)(2173,2174,2175,2176)
(2177,2178,2179,2180)(2181,2182,2183,2184)(2185,2186,2187,2188)(2189,2190,2191,2192)
(2193,2194,2195,2196)(2197,2198,2199,2200)(2201,2202,2203,2204)(2205,2206,2207,2208)
(2209,2210,2211,2212)(2213,2214,2215,2216)(2217,2218,2219,2220)(2221,2222,2223,2224)
(2225,2226,2227,2228)(2229,2230,2231,2232)(2233,2234,2235,2236)(2237,2238,2239,2240)
(2241,2242,2243,2244)(2245,2246,2247,2248)(2249,2250,2251,2252)(2253,2254,2255,2256)
(2257,2258,2259,2260)(2261,2262,2263,2264)(2265,2266,2267,2268)(2269,2270,2271,2272)
(2273,2274,2275,2276)(2277,2278,2279,2280)(2281,2282,2283,2284)(2285,2286,2287,2288)
(2289,2290,2291,2292)(2293,2294,2295,2296)(2297,2298,2299,2300)(2301,2302,2303,2304);


w:=
(   1,   2)(   3,   4)(   5,   6)(   7,   8)(   9,  10)(  11,  12)(  13,  14)(  15,  16)
(  17,  22)(  18,  21)(  19,  24)(  20,  23)(  25,  30)(  26,  29)(  27,  32)(  28,  31)
(  33,  38)(  34,  37)(  35,  40)(  36,  39)(  41,  46)(  42,  45)(  43,  48)(  44,  47)
(  49,  54)(  50,  53)(  51,  56)(  52,  55)(  57,  62)(  58,  61)(  59,  64)(  60,  63)
(  65,1625)(  66,1626)(  67,1627)(  68,1628)(  69,1398)(  70,1399)(  71,1400)(  72,1397)
(  73,1016)(  74,1013)(  75,1014)(  76,1015)(  77,2178)(  78,2179)(  79,2180)(  80,2177)
(  81, 952)(  82, 949)(  83, 950)(  84, 951)(  85, 534)(  86, 535)(  87, 536)(  88, 533)
(  89,1711)(  90,1712)(  91,1709)(  92,1710)(  93, 872)(  94, 869)(  95, 870)(  96, 871)
(  97, 175)(  98, 176)(  99, 173)( 100, 174)( 101,1422)( 102,1423)( 103,1424)( 104,1421)
( 105,1144)( 106,1141)( 107,1142)( 108,1143)( 109,1703)( 110,1704)( 111,1701)( 112,1702)
( 113, 678)( 114, 679)( 115, 680)( 116, 677)( 117,1717)( 118,1718)( 119,1719)( 120,1720)
( 121, 803)( 122, 804)( 123, 801)( 124, 802)( 125,1784)( 126,1781)( 127,1782)( 128,1783)
( 129,1471)( 130,1472)( 131,1469)( 132,1470)( 133, 345)( 134, 346)( 135, 347)( 136, 348)
( 137, 463)( 138, 464)( 139, 461)( 140, 462)( 141,1032)( 142,1029)( 143,1030)( 144,1031)
( 145,1434)( 146,1435)( 147,1436)( 148,1433)( 149,1694)( 150,1695)( 151,1696)( 152,1693)
( 153, 559)( 154, 560)( 155, 557)( 156, 558)( 157,1342)( 158,1343)( 159,1344)( 160,1341)
( 161,1064)( 162,1061)( 163,1062)( 164,1063)( 165,2056)( 166,2053)( 167,2054)( 168,2055)
( 169,1170)( 170,1171)( 171,1172)( 172,1169)( 177, 523)( 178, 524)( 179, 521)( 180, 522)
( 181,2199)( 182,2200)( 183,2197)( 184,2198)( 185, 299)( 186, 300)( 187, 297)( 188, 298)
( 189,1325)( 190,1326)( 191,1327)( 192,1328)( 193,1920)( 194,1917)( 195,1918)( 196,1919)
( 197, 443)( 198, 444)( 199, 441)( 200, 442)( 201,1079)( 202,1080)( 203,1077)( 204,1078)
( 205, 745)( 206, 746)( 207, 747)( 208, 748)( 209,1388)( 210,1385)( 211,1386)( 212,1387)
( 213, 631)( 214, 632)( 215, 629)( 216, 630)( 217,2157)( 218,2158)( 219,2159)( 220,2160)
( 221,1036)( 222,1033)( 223,1034)( 224,1035)( 225,1340)( 226,1337)( 227,1338)( 228,1339)
( 229, 452)( 230, 449)( 231, 450)( 232, 451)( 233,1724)( 234,1721)( 235,1722)( 236,1723)
( 237,2235)( 238,2236)( 239,2233)( 240,2234)( 241, 769)( 242, 770)( 243, 771)( 244, 772)
( 245, 783)( 246, 784)( 247, 781)( 248, 782)( 249,1593)( 250,1594)( 251,1595)( 252,1596)
( 253,1754)( 254,1755)( 255,1756)( 256,1753)( 257,1512)( 258,1509)( 259,1510)( 260,1511)
( 261, 818)( 262, 819)( 263, 820)( 264, 817)( 265,2242)( 266,2243)( 267,2244)( 268,2241)
( 269,1282)( 270,1283)( 271,1284)( 272,1281)( 273,1454)( 274,1455)( 275,1456)( 276,1453)
( 277,1850)( 278,1851)( 279,1852)( 280,1849)( 281,1426)( 282,1427)( 283,1428)( 284,1425)
( 285,2249)( 286,2250)( 287,2251)( 288,2252)( 289, 942)( 290, 943)( 291, 944)( 292, 941)
( 293, 809)( 294, 810)( 295, 811)( 296, 812)( 301,1912)( 302,1909)( 303,1910)( 304,1911)
( 305,1261)( 306,1262)( 307,1263)( 308,1264)( 309,2269)( 310,2270)( 311,2271)( 312,2272)
( 313, 590)( 314, 591)( 315, 592)( 316, 589)( 317, 431)( 318, 432)( 319, 429)( 320, 430)
( 321,2279)( 322,2280)( 323,2277)( 324,2278)( 325,1462)( 326,1463)( 327,1464)( 328,1461)
( 329,1808)( 330,1805)( 331,1806)( 332,1807)( 333,1528)( 334,1525)( 335,1526)( 336,1527)
( 337,1067)( 338,1068)( 339,1065)( 340,1066)( 341, 618)( 342, 619)( 343, 620)( 344, 617)
( 349,1111)( 350,1112)( 351,1109)( 352,1110)( 353,2167)( 354,2168)( 355,2165)( 356,2166)
( 357,1412)( 358,1409)( 359,1410)( 360,1411)( 361,2145)( 362,2146)( 363,2147)( 364,2148)
( 365,1056)( 366,1053)( 367,1054)( 368,1055)( 369, 657)( 370, 658)( 371, 659)( 372, 660)
( 373,1961)( 374,1962)( 375,1963)( 376,1964)( 377, 484)( 378, 481)( 379, 482)( 380, 483)
( 381,1944)( 382,1941)( 383,1942)( 384,1943)( 385, 531)( 386, 532)( 387, 529)( 388, 530)
( 389,1987)( 390,1988)( 391,1985)( 392,1986)( 393, 410)( 394, 411)( 395, 412)( 396, 409)
( 397,1869)( 398,1870)( 399,1871)( 400,1872)( 401, 732)( 402, 729)( 403, 730)( 404, 731)
( 405,2065)( 406,2066)( 407,2067)( 408,2068)( 413,1548)( 414,1545)( 415,1546)( 416,1547)
( 417,1670)( 418,1671)( 419,1672)( 420,1669)( 421, 777)( 422, 778)( 423, 779)( 424, 780)
( 425,1305)( 426,1306)( 427,1307)( 428,1308)( 433, 909)( 434, 910)( 435, 911)( 436, 912)
( 437,1876)( 438,1873)( 439,1874)( 440,1875)( 445,1524)( 446,1521)( 447,1522)( 448,1523)
( 453,1361)( 454,1362)( 455,1363)( 456,1364)( 457,1793)( 458,1794)( 459,1795)( 460,1796)
( 465,2039)( 466,2040)( 467,2037)( 468,2038)( 469,1879)( 470,1880)( 471,1877)( 472,1878)
( 473, 897)( 474, 898)( 475, 899)( 476, 900)( 477,1174)( 478,1175)( 479,1176)( 480,1173)
( 485,2023)( 486,2024)( 487,2021)( 488,2022)( 489, 750)( 490, 751)( 491, 752)( 492, 749)
( 493, 762)( 494, 763)( 495, 764)( 496, 761)( 497, 905)( 498, 906)( 499, 907)( 500, 908)
( 501,1113)( 502,1114)( 503,1115)( 504,1116)( 505,1812)( 506,1809)( 507,1810)( 508,1811)
( 509,2029)( 510,2030)( 511,2031)( 512,2032)( 513, 650)( 514, 651)( 515, 652)( 516, 649)
( 517,1931)( 518,1932)( 519,1929)( 520,1930)( 525,1298)( 526,1299)( 527,1300)( 528,1297)
( 537,1995)( 538,1996)( 539,1993)( 540,1994)( 541, 645)( 542, 646)( 543, 647)( 544, 648)
( 545, 794)( 546, 795)( 547, 796)( 548, 793)( 549,1960)( 550,1957)( 551,1958)( 552,1959)
( 553,1815)( 554,1816)( 555,1813)( 556,1814)( 561,1532)( 562,1529)( 563,1530)( 564,1531)
( 565,1757)( 566,1758)( 567,1759)( 568,1760)( 569, 837)( 570, 838)( 571, 839)( 572, 840)
( 573,1093)( 574,1094)( 575,1095)( 576,1096)( 577,1928)( 578,1925)( 579,1926)( 580,1927)
( 581,2026)( 582,2027)( 583,2028)( 584,2025)( 585, 880)( 586, 877)( 587, 878)( 588, 879)
( 593,1439)( 594,1440)( 595,1437)( 596,1438)( 597,1304)( 598,1301)( 599,1302)( 600,1303)
( 601,1971)( 602,1972)( 603,1969)( 604,1970)( 605,1415)( 606,1416)( 607,1413)( 608,1414)
( 609, 936)( 610, 933)( 611, 934)( 612, 935)( 613,1227)( 614,1228)( 615,1225)( 616,1226)
( 621,1487)( 622,1488)( 623,1485)( 624,1486)( 625,2011)( 626,2012)( 627,2009)( 628,2010)
( 633,1921)( 634,1922)( 635,1923)( 636,1924)( 637,1252)( 638,1249)( 639,1250)( 640,1251)
( 641,1553)( 642,1554)( 643,1555)( 644,1556)( 653,2169)( 654,2170)( 655,2171)( 656,2172)
( 661,1254)( 662,1255)( 663,1256)( 664,1253)( 665, 970)( 666, 971)( 667, 972)( 668, 969)
( 669,1126)( 670,1127)( 671,1128)( 672,1125)( 673,1473)( 674,1474)( 675,1475)( 676,1476)
( 681,1896)( 682,1893)( 683,1894)( 684,1895)( 685,2044)( 686,2041)( 687,2042)( 688,2043)
( 689,2240)( 690,2237)( 691,2238)( 692,2239)( 693, 996)( 694, 993)( 695, 994)( 696, 995)
( 697,1122)( 698,1123)( 699,1124)( 700,1121)( 701,2049)( 702,2050)( 703,2051)( 704,2052)
( 705, 757)( 706, 758)( 707, 759)( 708, 760)( 709,1482)( 710,1483)( 711,1484)( 712,1481)
( 713,1620)( 714,1617)( 715,1618)( 716,1619)( 717,1824)( 718,1821)( 719,1822)( 720,1823)
( 721,1086)( 722,1087)( 723,1088)( 724,1085)( 725,1857)( 726,1858)( 727,1859)( 728,1860)
( 733,1027)( 734,1028)( 735,1025)( 736,1026)( 737,2286)( 738,2287)( 739,2288)( 740,2285)
( 741,1610)( 742,1611)( 743,1612)( 744,1609)( 753, 798)( 754, 799)( 755, 800)( 756, 797)
( 765,1402)( 766,1403)( 767,1404)( 768,1401)( 773,2075)( 774,2076)( 775,2073)( 776,2074)
( 785,2005)( 786,2006)( 787,2007)( 788,2008)( 789,1897)( 790,1898)( 791,1899)( 792,1900)
( 805,2181)( 806,2182)( 807,2183)( 808,2184)( 813,1901)( 814,1902)( 815,1903)( 816,1904)
( 821,1294)( 822,1295)( 823,1296)( 824,1293)( 825,1490)( 826,1491)( 827,1492)( 828,1489)
( 829, 968)( 830, 965)( 831, 966)( 832, 967)( 833,1508)( 834,1505)( 835,1506)( 836,1507)
( 841,1131)( 842,1132)( 843,1129)( 844,1130)( 845,1976)( 846,1973)( 847,1974)( 848,1975)
( 849,1716)( 850,1713)( 851,1714)( 852,1715)( 853,1828)( 854,1825)( 855,1826)( 856,1827)
( 857,1707)( 858,1708)( 859,1705)( 860,1706)( 861,2213)( 862,2214)( 863,2215)( 864,2216)
( 865,2258)( 866,2259)( 867,2260)( 868,2257)( 873,1100)( 874,1097)( 875,1098)( 876,1099)
( 881,1072)( 882,1069)( 883,1070)( 884,1071)( 885,1467)( 886,1468)( 887,1465)( 888,1466)
( 889,1503)( 890,1504)( 891,1501)( 892,1502)( 893,2020)( 894,2017)( 895,2018)( 896,2019)
( 901, 988)( 902, 985)( 903, 986)( 904, 987)( 913,1989)( 914,1990)( 915,1991)( 916,1992)
( 917,1746)( 918,1747)( 919,1748)( 920,1745)( 921,1647)( 922,1648)( 923,1645)( 924,1646)
( 925,1309)( 926,1310)( 927,1311)( 928,1312)( 929,1539)( 930,1540)( 931,1537)( 932,1538)
( 937,2033)( 938,2034)( 939,2035)( 940,2036)( 945,1495)( 946,1496)( 947,1493)( 948,1494)
( 953,1478)( 954,1479)( 955,1480)( 956,1477)( 957,2188)( 958,2185)( 959,2186)( 960,2187)
( 961,2294)( 962,2295)( 963,2296)( 964,2293)( 973,1200)( 974,1197)( 975,1198)( 976,1199)
( 977,2100)( 978,2097)( 979,2098)( 980,2099)( 981,1076)( 982,1073)( 983,1074)( 984,1075)
( 989,1908)( 990,1905)( 991,1906)( 992,1907)( 997,1195)( 998,1196)( 999,1193)(1000,1194)
(1001,1586)(1002,1587)(1003,1588)(1004,1585)(1005,2230)(1006,2231)(1007,2232)(1008,2229)
(1009,1601)(1010,1602)(1011,1603)(1012,1604)(1017,1201)(1018,1202)(1019,1203)(1020,1204)
(1021,1165)(1022,1166)(1023,1167)(1024,1168)(1037,2218)(1038,2219)(1039,2220)(1040,2217)
(1041,1863)(1042,1864)(1043,1861)(1044,1862)(1045,2254)(1046,2255)(1047,2256)(1048,2253)
(1049,2176)(1050,2173)(1051,2174)(1052,2175)(1057,1575)(1058,1576)(1059,1573)(1060,1574)
(1081,1140)(1082,1137)(1083,1138)(1084,1139)(1089,1380)(1090,1377)(1091,1378)(1092,1379)
(1101,1321)(1102,1322)(1103,1323)(1104,1324)(1105,1682)(1106,1683)(1107,1684)(1108,1681)
(1117,1685)(1118,1686)(1119,1687)(1120,1688)(1133,1751)(1134,1752)(1135,1749)(1136,1750)
(1145,1636)(1146,1633)(1147,1634)(1148,1635)(1149,1290)(1150,1291)(1151,1292)(1152,1289)
(1153,1887)(1154,1888)(1155,1885)(1156,1886)(1157,1374)(1158,1375)(1159,1376)(1160,1373)
(1161,1790)(1162,1791)(1163,1792)(1164,1789)(1177,1457)(1178,1458)(1179,1459)(1180,1460)
(1181,2190)(1182,2191)(1183,2192)(1184,2189)(1185,2142)(1186,2143)(1187,2144)(1188,2141)
(1189,2088)(1190,2085)(1191,2086)(1192,2087)(1205,1736)(1206,1733)(1207,1734)(1208,1735)
(1209,2058)(1210,2059)(1211,2060)(1212,2057)(1213,1583)(1214,1584)(1215,1581)(1216,1582)
(1217,2014)(1218,2015)(1219,2016)(1220,2013)(1221,1984)(1222,1981)(1223,1982)(1224,1983)
(1229,2281)(1230,2282)(1231,2283)(1232,2284)(1233,1769)(1234,1770)(1235,1771)(1236,1772)
(1237,1544)(1238,1541)(1239,1542)(1240,1543)(1241,1741)(1242,1742)(1243,1743)(1244,1744)
(1245,1829)(1246,1830)(1247,1831)(1248,1832)(1257,1448)(1258,1445)(1259,1446)(1260,1447)
(1265,1866)(1266,1867)(1267,1868)(1268,1865)(1269,1933)(1270,1934)(1271,1935)(1272,1936)
(1273,1967)(1274,1968)(1275,1965)(1276,1966)(1277,1390)(1278,1391)(1279,1392)(1280,1389)
(1285,1843)(1286,1844)(1287,1841)(1288,1842)(1313,2211)(1314,2212)(1315,2209)(1316,2210)
(1317,2001)(1318,2002)(1319,2003)(1320,2004)(1329,1558)(1330,1559)(1331,1560)(1332,1557)
(1333,2081)(1334,2082)(1335,2083)(1336,2084)(1345,2129)(1346,2130)(1347,2131)(1348,2132)
(1349,1661)(1350,1662)(1351,1663)(1352,1664)(1353,1451)(1354,1452)(1355,1449)(1356,1450)
(1357,1551)(1358,1552)(1359,1549)(1360,1550)(1365,1534)(1366,1535)(1367,1536)(1368,1533)
(1369,1442)(1370,1443)(1371,1444)(1372,1441)(1381,2103)(1382,2104)(1383,2101)(1384,2102)
(1393,1608)(1394,1605)(1395,1606)(1396,1607)(1405,1676)(1406,1673)(1407,1674)(1408,1675)
(1417,1999)(1418,2000)(1419,1997)(1420,1998)(1429,1892)(1430,1889)(1431,1890)(1432,1891)
(1497,1776)(1498,1773)(1499,1774)(1500,1775)(1513,1845)(1514,1846)(1515,1847)(1516,1848)
(1517,1656)(1518,1653)(1519,1654)(1520,1655)(1561,2045)(1562,2046)(1563,2047)(1564,2048)
(1565,2071)(1566,2072)(1567,2069)(1568,2070)(1569,1913)(1570,1914)(1571,1915)(1572,1916)
(1577,1838)(1578,1839)(1579,1840)(1580,1837)(1589,2276)(1590,2273)(1591,2274)(1592,2275)
(1597,1833)(1598,1834)(1599,1835)(1600,1836)(1613,1689)(1614,1690)(1615,1691)(1616,1692)
(1621,2118)(1622,2119)(1623,2120)(1624,2117)(1629,1659)(1630,1660)(1631,1657)(1632,1658)
(1637,2106)(1638,2107)(1639,2108)(1640,2105)(1641,2194)(1642,2195)(1643,2196)(1644,2193)
(1649,1856)(1650,1853)(1651,1854)(1652,1855)(1665,1698)(1666,1699)(1667,1700)(1668,1697)
(1677,2201)(1678,2202)(1679,2203)(1680,2204)(1725,2150)(1726,2151)(1727,2152)(1728,2149)
(1729,2289)(1730,2290)(1731,2291)(1732,2292)(1737,1817)(1738,1818)(1739,1819)(1740,1820)
(1761,2111)(1762,2112)(1763,2109)(1764,2110)(1765,2138)(1766,2139)(1767,2140)(1768,2137)
(1777,2128)(1778,2125)(1779,2126)(1780,2127)(1785,2093)(1786,2094)(1787,2095)(1788,2096)
(1797,2164)(1798,2161)(1799,2162)(1800,2163)(1801,2091)(1802,2092)(1803,2089)(1804,2090)
(1881,2134)(1882,2135)(1883,2136)(1884,2133)(1937,2080)(1938,2077)(1939,2078)(1940,2079)
(1945,1951)(1946,1952)(1947,1949)(1948,1950)(1953,2114)(1954,2115)(1955,2116)(1956,2113)
(1977,2224)(1978,2221)(1979,2222)(1980,2223)(2061,2263)(2062,2264)(2063,2261)(2064,2262)
(2121,2300)(2122,2297)(2123,2298)(2124,2299)(2153,2302)(2154,2303)(2155,2304)(2156,2301)
(2205,2266)(2206,2267)(2207,2268)(2208,2265)(2225,2245)(2226,2246)(2227,2247)(2228,2248);
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Proof of result

Submitted by Paul Timmons on Sun, 05/09/2021 - 10:00.

Someone PM'd me to request actual computational details and proof of the result so here goes.

Prerequisites: MAF downloaded on Windows PC with at least 8Gig RAM
Create a text file Tits_nice in the C:\maf-v2.2.1-windows-x86\examples\automatic or similar subdirectory as follows 

_RWS := rec
(
  isRWS := true,
  ordering := "shortlex",
  generatorOrder := [a,A,b,B],
  inverses := [A,a,B,b],
  equations :=
  [
    [a^2*b^4, IdWord],
# Following palindromic on LHS and RHS combines (ab)^13 and (ababaBaB)^2 relators
    [b*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b, a*b^2*a*B*a*b*a*b*a*B*a*b^2*a],
    [(a*b*a*B)^2*(a*b)^3*a*b^2*a*B*a*b^2*a*b*a*b^2*(a*B)^3*(a*b*a*B)^2, IdWord]
  ]
);


Execute the following command from the COMMAND prompt 
C:\maf-v2.2.1-windows-x86\examples\automatic>..\..\bin\automata.exe -nowd Tits_nice


After approx 2.5 hours out the following output should be produced 
B^3=a^2*b
b^3=A^2*B
Looking for new equations using Knuth Bendix process
Expanding Nodes at LOD 4 (96 of 1581). Size L/R 15/15
Nodes:L0=48955,L1=7534,L2/3=25,Bad=0,Depth=22,Acc=17
Limits:E=32/41,S/T/P/D=39/42/43/51,L=17/17/22,O=32,C=52,F=39
Pool: Best LHS 22/43 Best RHS 21/43
Equations:Total=14465, Using=7534, Visible=7559, Pool=6906.
a^17=A^15
A^16=a^16
Expanding Nodes at LOD 4 (1057 of 1583). Size L/R 13/13
Nodes:L0=58636,L1=9884,L2/3=37,Bad=0,Depth=22,Acc=17
Limits:E=32/41,S/T/P/D=39/42/43/51,L=17/17/22,O=32,C=52,F=39
Pool: Best LHS 22/43 Best RHS 21/43
Equations:Total=30593, Using=9884, Visible=9921, Pool=20672.
Heap use so far: In use 7936242 Reserved 12582936, Utilisation 63%
Examining pool (5132 of 26017) (2091 adopted)
Nodes:L0=118609,L1=16279,L2/3=42,Bad=0,Depth=25,Acc=19
Limits:E=33/41,S/T/P/D=44/48/50/58,L=17/17/22,O=32,C=52,F=39
Pool: Best LHS 25/50 Best RHS 25/50
Equations:Total=37399, Using=16279, Visible=16321, Pool=193.
Examining pool (10217 of 26017) (3283 adopted)
Nodes:L0=152230,L1=20431,L2/3=53,Bad=0,Depth=25,Acc=18
Limits:E=33/41,S/T/P/D=44/48/49/57,L=17/17/22,O=32,C=52,F=39
Pool: Best LHS 25/49 Best RHS 24/49
Equations:Total=38097, Using=20431, Visible=20484, Pool=1813.
Examining pool (21294 of 26017) (7339 adopted)
Nodes:L0=192666,L1=26184,L2/3=138,Bad=0,Depth=25,Acc=19
Limits:E=33/41,S/T/P/D=44/48/49/57,L=17/17/22,O=32,C=52,F=39
Pool: Best LHS 25/49 Best RHS 24/49

...
...
...

All possible primary equations have been deduced
Examining nodes (6819585 of 17461413)
Examining nodes (15232513 of 17461413)
Building index automaton (289766 of 1062630 nodes to do). 33136 equations
Building index automaton (560298 of 2210986 nodes to do). 84384 equations
Building index automaton (763342 of 3192526 nodes to do). 138441 equations
...
...
...
Building index automaton (122834 of 10974418 nodes to do). 2991895 equations
Heap use so far: In use 1338183766 Reserved 1354730624, Utilisation 98%
Examining nodes (1999873 of 17461413)
Examining nodes (9601537 of 17461413)
Heap use so far: In use 1354914417 Reserved 1369500648, Utilisation 98%
Creating FSA states (8010496 of 11016660 done)
Index automaton has 11016660 states and 3149772 equations
Building word-acceptor from confluent RWS
Heap use so far: In use 1412329778 Reserved 1615004480, Utilisation 87%
FSA pre-minimise pass 0 with 11016660 categories.(Checking 11016660 states)
FSA pre-minimise pass 1 with 9155031 categories.(Checking 9155031 states)
FSA pre-minimise pass 2 with 7742982 categories.(Checking 7742982 states)
FSA pre-minimise pass 3 with 6577162 categories.(Checking 6577162 states)
FSA pre-minimise pass 4 with 5569299 categories.(Checking 5569299 states)
FSA pre-minimise pass 5 with 4684746 categories.(Checking 4684746 states)
FSA pre-minimise pass 6 with 3940771 categories.(Checking 3940771 states)
FSA pre-minimise pass 7 with 3412996 categories.(Checking 3412996 states)
FSA minimise pass 0 with 0 states so far. (Checking 3129669 states)
FSA minimise pass 2 with 27 states so far. (Checking 3129669 states)
FSA minimise pass 4 with 2497 states so far. (Checking 3129669 states)
FSA minimise pass 6 with 801721 states so far. (Checking 3129669 states)
FSA minimise pass 7 with 1716952 states so far. (Checking 3129669 states)
FSA minimise pass 8 with 2350330 states so far. (Checking 3129669 states)
FSA minimise pass 9 with 2683156 states so far. (Checking 3129669 states)
FSA minimise pass 10 with 2839332 states so far. (Checking 3129669 states)
FSA minimise pass 11 with 2912087 states so far. (Checking 3129669 states)
FSA minimise pass 12 with 2945878 states so far. (Checking 3129669 states)
FSA minimise pass 13 with 2961343 states so far. (Checking 3129669 states)
FSA minimise pass 14 with 2968187 states so far. (Checking 3129669 states)
FSA minimise pass 15 with 2970995 states so far. (Checking 3129669 states)
FSA minimise pass 16 with 2972141 states so far. (Checking 3129669 states)
FSA minimise pass 17 with 2972579 states so far. (Checking 3129669 states)
FSA minimise pass 18 with 2972750 states so far. (Checking 3129669 states)
FSA minimise pass 19 with 2972829 states so far. (Checking 3129669 states)
FSA minimise pass 20 with 2972864 states so far. (Checking 3129669 states)
FSA minimise pass 21 with 2972876 states so far. (Checking 3129669 states)
FSA minimise pass 22 with 2972885 states so far. (Checking 3129669 states)
FSA minimise pass 23 with 2972889 states so far. (Checking 3129669 states)
FSA minimise pass 24 with 2972891 states so far. (Checking 3129669 states)
FSA minimise pass 25 with 2972893 states so far. (Checking 3129669 states)
Accepted language contains 17971200 words. Word-acceptor has 2972893 states
Building L1 acceptor
Building L1 acceptor state 1843200 (583445 of 2426645 to do). Depth 32
Building L1 acceptor state 3278848 (689628 of 3968476 to do). Depth 34
Building L1 acceptor state 4460288 (466166 of 4926454 to do). Depth 36
Heap use so far: In use 1400867566 Reserved 1541076960, Utilisation 90%
FSA minimise pass 0 with 0 states so far. (Checking 4759989 states)
FSA minimise pass 2 with 78 states so far. (Checking 4759989 states)
FSA minimise pass 4 with 26148 states so far. (Checking 4759989 states)
FSA minimise pass 5 with 359101 states so far. (Checking 4759989 states)
FSA minimise pass 6 with 1145403 states so far. (Checking 4759989 states)
FSA minimise pass 7 with 1904463 states so far. (Checking 4759989 states)
FSA minimise pass 8 with 2348338 states so far. (Checking 4759989 states)
FSA minimise pass 9 with 2551236 states so far. (Checking 4759989 states)
FSA minimise pass 10 with 2631998 states so far. (Checking 4759989 states)
FSA minimise pass 11 with 2661171 states so far. (Checking 4759989 states)
FSA minimise pass 12 with 2671263 states so far. (Checking 4759989 states)
FSA minimise pass 13 with 2674784 states so far. (Checking 4759989 states)
FSA minimise pass 14 with 2676256 states so far. (Checking 4759989 states)
FSA minimise pass 15 with 2676865 states so far. (Checking 4759989 states)
FSA minimise pass 16 with 2677069 states so far. (Checking 4759989 states)
FSA minimise pass 17 with 2677131 states so far. (Checking 4759989 states)
FSA minimise pass 18 with 2677149 states so far. (Checking 4759989 states)
FSA minimise pass 19 with 2677158 states so far. (Checking 4759989 states)
FSA minimise pass 20 with 2677162 states so far. (Checking 4759989 states)
The L1 language contains 3149772 words
The L1 acceptor has 2677162 states.
Examining nodes (1 of 17461413)
Examining nodes (6435585 of 17461413)
Examining nodes (13662721 of 17461413)
Building index automaton (164220 of 573820 nodes to do). 20167 equations
Building index automaton (424729 of 1612057 nodes to do). 71346 equations
...
...
...
Building index automaton (77235 of 11178419 nodes to do). 3215586 equations
Heap use so far: In use 1739865187 Reserved 1779317120, Utilisation 97%
Examining nodes (3687425 of 17461413)
Examining nodes (11354113 of 17461413)
Heap use so far: In use 1762952326 Reserved 1801835544, Utilisation 97%
Creating FSA states (9162240 of 11200823 done)
Index automaton has 11200823 states and 3312984 equations
Elapsed time 9129


To obtain the antipodes of length 49 type the following 


C:\maf-v2.2.1-windows-x86\examples\automatic>..\..\bin\fsaenumerate 49 49 Tits_nice.wa
Heap use so far: In use 23720349 Reserved 24474528, Utilisation 96%
Reading file. 1626028 lines read so far

C:\maf-v2.2.1-windows-x86\examples\automatic>type Tits_nice.wa.enumerate
Tits_nice.wa :=
[
  a*b^2*a*b*a*b*a*b^2*a*b^2*a*B*a*b*a*b^2*a*b*a*B*a*b*a*b^2*a*b*a*b*a*B*a*b*a*b^2*a*b*a*B*a*b^2*a*b,
  a*b^2*a*b*a*b*a*b^2*a*b^2*a*B*a*b*a*b^2*a*b*a*B*a*b*a*b^2*a*b*a*b*a*B*a*b*a*b^2*a*b*a*B*a*b^2*a*B
];

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