Abstract group 5389606560.a: $\PSL(2,2209)$

• Label: 5389606560.a
• Order: \( 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $2210$
• Trans deg.: $2210$
• Rank: $2$

Group information

Description:	$\PSL(2,2209)$
Order:	\(5389606560\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47^{2} \)
Exponent:	\(57336240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47 \)
Automorphism group:	Group of order \(21558426240\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 47^{2} \)
Composition factors:	$\PSL(2,2209)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	8	12	13	16	17	23	24	46	47	48	65	69	85	92	138	184	221	276	368	552	1104	1105
Elements	1	2440945	4881890	4881890	9754944	4881890	9763780	9763780	29264832	19527560	39019776	53700790	19527560	53700790	4879680	39055120	117059328	107401580	156079104	107401580	107401580	214803160	468237312	214803160	429606320	429606320	859212640	1872949248	5389606560
Conjugacy classes	1	1	1	1	2	1	2	2	6	4	8	11	4	11	2	8	24	22	32	22	22	44	96	44	88	88	176	384	1107
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	29
Autjugacy classes	1	1	1	1	1	1	2	2	3	4	4	11	4	11	1	8	12	11	16	11	11	22	48	22	44	44	88	192	577

Minimal presentations

Permutation degree:	$2210$
Transitive degree:	$2210$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1105	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSL(2,2209)$, $\PSU(2,2209)$, $\Omega(3,2209)$, $\OmegaMinus(4,47)$, $\POmega(3,2209)$, $\POmegaMinus(4,47)$
Permutation group:	Degree $2210$ $\langle(3,1477,1198,1722,1405,1220,2060,1793,2070,2005,760,975,360,1672,446,558,1503,631,1214,145,459,1264,1774,103,923,485,1170,515,1303,1648,578,777,2208,1710,1397,685,1639,1002,2207,1624,53,1435,1555,1395,1761,1514,21,1741,783,1985,899,2202,1484,1070,1037,356,731,166,844,1191,2148,547,684,1675,2201,1429,1988,1991,211,386,642,1418,234,1487,1685,227,798,1122,2019,322,1390,1372,1468,1040,741,1573,434,1544,1402,210,1140,1004,1433,159,479,651,1543,1731,290,1380,419,246,1569,748,44,38,180,365,690,1237,1873,579,1538,369,1434,958,663,221,1050,1239,1928,1185,1283,78,2163,493,2170,952,570,162,1259,2069,2197,1307,383,303,374,2134,2062,1673,1933,350,502,1944,1782,67,1958,597,1253,1272,628,1602,608,55,206,1971,839,2141,414,1300,2004,1403,1925,13,1957,97,1766,458,1216,645,980,861,1189,49,1577,308,951,836,101,789,662,406,584,396,1860,1814,564,2027,1225,1489,1978,29,495,1865,1383,1109,1976,66,870,1033,1071,1567,1941,1048,1696,1136,953,1208,412,473,428,926,1816,1831,93,422,1940,1250,1061,1281,1316,2196,711,568,2108,270,465,739,178,832,165,405,1699,1628,2171,1021,1475,1505,84,1772,1980,1689,1880,1511,934,1843,1081,753,114,510,313,792,301,2156,1935,1604,1126,342,600,531,1045,935,785,1644,1291,679,1293,2006,865,1029,1382,264,591,1744,2065,1924,1344,1950,1328,1267,129,435,163,292,2100,120,1055,907,1407,1975,602,1466,181,1961,1691,2089,1288,1438,1017,203,929,1724,704,2013,814,919,1277,204,732,1039,2144,1008,2091,604,2085,223,1759,1849,326,530,466,2126,371,1736,372,1265,1462,1018,1186,323,392,193,961,1023,436,429,1590,1340,1959,1123,834,2123,329,1467,424,2045,915,175,1411,102,2026,379,1197,167,250,544,1932,1911,1342,1610,432,2066,790,1456,2016,2096,1150,1592,561,1826,781,40,931,1530,1982,31,1394,1886,1792,2124,2092,2142,1650,1448,1330,1068,18,1213,647,109,1862,260,1094,718,1825,968,1341,908,306,1868,76,2162,1357,743,28,77,2023,825,156,573,1716,36,1945,1304,983,852,1584,1447,686,1690,468,133,534,696,1697,1086,991,2167,1539,1128,423,1463,1107,1545,667,1301,1310,1775,638,401,8,1751,2014,595,99,1549,1169,134,776,1837,276,701,173,427,1867,1852,33,648,677,1846,1537,130,945,1413,2103,1454,247,875,2037,619,1559,262,857,1642,1201,1093,889,2101,1513,2107,1431,607,1319,1449,1494,1444,1956,1874,575,986,1053,258,622,545,672,842,964,1914,1798,1756,665,259,1441,2179,2058,1251,2030,1946,1051,1228,837,1224,277,826,1997,1815,1057,727,1116,400,2059,1836,829,1054,709,1637,287,1802,1719,1581,762,1235,2012,146,355,627,1345,1193,1248,1630,150,2194,1769,888,1187,1207,1903,1764,1271,2178,1247,1981,879,1773,244,1231,136,567,2018,1243,467,1521,896,1336,283,1105,1346,1901,1046,471,2001,4,1148,316,533,185,274,586,1856,1833,1535,131,1309,2073,858,1153,487,1524,218,1254,344,562,1717,1367,304,202,1686,1593,1855,981,2181,666,137,9,300,898,1440,2139,470,152,1118,1289,47,450,1393,1066,1795,1515,1647,230,2187,2086,1111,240,286,626,577,474,876,499,1183,215,895,1677,1134,10,539,788,559,483,594,1671,1768,1490,967,317,1970,1962,1364,1917,284,335,1557,381,1075,687,52,1760,1064,75,1706,1236,1866,475,1701,2024,1845,1458,248,1556,1450,1113,1335,1137,160,943,794,555,408,2078,799,625,100,2193,1540,691,618,1155,476,1312,2155,61,1723,59,892,1276,850,1347,1370,2157,1621,694,1103,650,1947,1725,2132,2198,407,1875,2159,824,224,835,1410,660,118,891,1001,481,1084,115,282,1417,1934,1217,2105,48,639,363,1531,525,1510,730,149,927,142,1659,1750,721,882,410,509,1041,624,184,1942,1822,1275,23,451,1359,416,1473,2106,1010,838,1242,1664,556,712,1469,716,1695,1206,2118,1877,1368,1580,172,1676,886,1653,1313,1392,635,1333,1488,1987,1179,480,236,1121,957,1895,409,1047,1885,1906,1400,1098,359,864,1597,80,245,599,1038,1838,1583,1820,293,325,1752,2119,330,774,1564,1339,526,1404,249,501,198,1025,417,998,1908,343,349,1157,1114,158,615,1727,933,315,311,1013,2183,714,956,1135,196,990,2186,1052,1014,484,552,123,767,1670,805,464,1171,1442,1234,2087,2074,992,261,815,1840,385,402,1996,1365,688,1715,728,697,1532,588,884,448,1823,1662,1326,1829,1694,86,2039,695,1295,847,557,693,1117,655,1603,179,1299,1323,2044,1177,1600,1258,521,1554,1800,572,309,1426,580,352,2128,1072,1178,1445,853,1776,2000,1493,331,24,1031,1362,395,1916,192,529,1693,1396,590,984,598,1125,939,614,354,1631,298,1508,2121,1292,733,1960,2025,1667,1218,1412,1074,241,1721,770,671,170,1617,1422,291,1787,917,294,1607,987,1552,1067,433,511,500,994,1373,752,1308,68,2072,442,1533,581,634,1080,1358,157,1507,1948,64,2176,2117,822,1999,82,25,2206,1455,2174,1059,403,1452,1872,973,1351,1889,593,828,2115,1678,1835,2088,916,1665,737,460,60,1194,1062,1709,1498,1302,70,1681,1332,1984,425,860,617,1851,1479,807,1082,2140,1457,1500,219,1612,1974,191,928,1320,2081,1993,1608,404,2150,1892,1496,742,205,1269,612,2094,505,1130,1881,275,41,1876,394,182,2191,1879,950,1626,455,902,1899,812,553,1506,1003,947,138,2052,1459,749,1757,1386,804,641,653,1273,83,2125,2199,1783,478,1847,1049,1354,1918,969,2102,1112,506,45,652,1202,972,1893,1009,1350,2122,104,177,722,513,443,2049,550,1124,1416,1570,793,508,735,116,62,171,1894,389,786,1069,1032,1576) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as equivalence classes (represented by square brackets) of matrices in $\SL(2,2209)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 5820432553 subgroups in 107 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\PSL(2,2209)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$
47-Sylow subgroup:	$P_{ 47 } \simeq$ $C_{47}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 5389606560 (order at least 1) are shown, as well as all normal subgroups of any index.

Order 5389606560: $\PSL(2,2209)$

Order 2438736: $C_{47}^2.C_{276}.C_4$

Order 1219368: $C_{47}^2:C_{552}$

Order 812912: $C_{47}^2:C_{368}$

Order 609684: $C_{47}^2:C_{276}$

Order 406456: $C_{47}^2:C_{184}$

Order 304842: $C_{47}^2:C_{138}$

Order 203228: $C_{47}^2:C_{92}$

Order 152421: $C_{47}^2:C_{69}$

Order 106032: $C_{47}^2:C_{48}$

Order 103776: $\PGL(2,47)$ x 2

Order 101614: $C_{47}^2:C_{46}$

Order 53016: $C_{47}^2:C_{24}$

Order 51888: $\PSL(2,47)$ x 2

Order 50807: $C_{47}^2:C_{23}$

Order 35344: $C_{47}^2:C_{16}$

Order 26508: $C_{47}^2:C_{12}$

Order 17672: $C_{47}^2:C_8$

Order 13254: $C_{47}^2:C_6$

Order 8836: $C_{47}^2:C_4$

Order 6627: $C_{47}^2:C_3$

Order 4418: $C_{47}^2:C_2$

Order 2210: $D_{1105}$

Order 2209: $C_{47}^2$

Order 2208: $D_{1104}$

Order 2162: $F_{47}$ x 2

Order 1105: $C_{1105}$

Order 1104: $D_{552}$ x 2, $C_{1104}$

Order 1081: $C_{47}:C_{23}$ x 2

Order 736: $D_{368}$

Order 552: $D_{276}$ x 2, $C_{552}$

Order 442: $D_{221}$

Order 368: $D_{184}$ x 2, $C_{368}$

Order 276: $D_{138}$ x 2, $C_{276}$

Order 221: $C_{221}$

Order 184: $D_{92}$ x 2, $C_{184}$

Order 170: $D_{85}$

Order 138: $D_{69}$ x 2, $C_{138}$

Order 130: $D_{65}$

Order 96: $D_{48}$

Order 94: $D_{47}$ x 2

Order 92: $D_{46}$ x 2, $C_{92}$

Order 85: $C_{85}$

Order 69: $C_{69}$

Order 65: $C_{65}$

Order 60: $A_5$ x 2

Order 48: $D_{24}$ x 2, $C_{48}$

Order 47: $C_{47}$ x 2

Order 46: $D_{23}$ x 2, $C_{46}$

Order 34: $D_{17}$

Order 32: $D_{16}$

Order 26: $D_{13}$

Order 24: $D_{12}$ x 2, $S_4$ x 2, $C_{24}$

Order 23: $C_{23}$

Order 17: $C_{17}$

Order 16: $D_8$ x 2, $C_{16}$

Order 13: $C_{13}$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_{12}$

Order 10: $D_5$

Order 8: $D_4$ x 2, $C_8$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 2, $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 5389606560: $\PSL(2,2209)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,2209)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $1107 \times 1107$ character table is not available for this group.

Rational character table

The $29 \times 29$ rational character table is not available for this group.