Abstract group 3160680600.a: $\PSL(2,1849)$

• Label: 3160680600.a
• Order: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $1850$
• Trans deg.: $1850$
• Rank: $2$

Group information

Description:	$\PSL(2,1849)$
Order:	\(3160680600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \)
Exponent:	\(36752100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43 \)
Automorphism group:	Group of order \(12642722400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \)
Composition factors:	$\PSL(2,1849)$
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	7	11	12	14	21	22	25	28	33	37	42	43	44	66	77	84	132	154	185	231	308	462	924	925
Elements	1	1710325	3420650	3420650	6833904	3420650	10261950	17103250	6841300	10261950	20523900	17103250	34169520	20523900	34206500	61505136	20523900	3418800	34206500	34206500	102619500	41047800	68413000	102619500	246020544	205239000	205239000	205239000	410478000	1230102720	3160680600
Conjugacy classes	1	1	1	1	2	1	3	5	2	3	6	5	10	6	10	18	6	2	10	10	30	12	20	30	72	60	60	60	120	360	927
Divisions	1	1	1	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	31
Autjugacy classes	1	1	1	1	1	1	3	5	1	3	6	5	5	3	5	9	6	1	10	5	15	6	10	15	36	30	30	30	60	180	485

Minimal presentations

Permutation degree:	$1850$
Transitive degree:	$1850$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	925	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSL(2,1849)$, $\PSU(2,1849)$, $\Omega(3,1849)$, $\OmegaMinus(4,43)$, $\POmega(3,1849)$, $\POmegaMinus(4,43)$
Permutation group:	Degree $1850$ $\langle(3,1599,976,371,163,1149,751,617,1464,983,398,369,712,370,246,835,1073,1479,1392,889,1175,449,1248,427,768,86,1089,404,71,1427,1213,1622,101,867,211,718,1746,812,177,646,1678,1299,1684,1101,796,85,214,480,990,507,739,1832,714,803,1082,1025,1047,801,668,357,1811,1714,446,1569,151,1217,1147,1508,627,15,612,1302,255,1582,30,1168,562,1814,1751,131,355,150,587,1385,467,969,1121,1769,729,928,281,630,1534,821,206,516,1773,1336,56,115,1480,880,1580,1242,1661,1828,1658,421,1250,1557,1058,1158,298,65,1630,836,1078,623,552,1246,956,1573,1612,735,859,1344,989,100,406,395,1081,1665,1394,1409,661,873,1070,665,1556,1466,692,1524,1231,950,38,327,831,1307,982,165,72,1251,1187,833,294,900,1145,51,413,217,360,1764,738,790,260,1600,844,249,388,305,1294,1671,1013,1539,1650,282,526,1490,759,381,1563,70,664,875,1319,1748,1786,1597,1574,199,1536,1616,46,784,887,1408,1331,354,225,1295,1743,1796,881,506,1813,593,452,486,143,794,17,1652,955,1519,705,724,1694,1224,1170,109,1328,529,585,1091,1716,696,1263,845,1035,1015,841,1297,1843,910,655,1435,1611,432,800,934,29,1210,1673,1273,1042,1613,311,93,648,487,1174,478,1750,366,472,342,451,1119,804,688,1126,1656,1360,1092,181,1804,1120,645,570,1816,227,382,805,601,674,1762,168,1203,194,604,1339,548,134,1845,1430,232,1735,1315,1068,1096,830,1003,1837,1686,1335,5,1031,1016,306,654,420,1619,1579,48,1771,1627,778,1434,1275,128,403,64,458,415,412,659,1540,295,288,1633,1310,1844,1040,637,753,827,157,1544,222,1270,1756,998,1083,53,1798,104,1111,1554,261,663,625,1051,457,899,890,1834,154,149,937,968,1244,1525,606,239,135,1271,895,1522,379,1729,1692,59,1790,566,263,385,1308,1799,924,896,528,975,1660,1136,1629,913,624,949,709,1355,1503,1538,450,1351,1413,1400,633,1663,270,1494,523,1810,1106,1689,854,133,392,1389,1801,1332,519,791,608,1342,1098,122,340,1518,1014,581,1491,1507,1338,647,437,431,746,565,471,615,861,829,691,1056,1361,1589,865,24,888,1728,1608,1406,905,561,687,1021,1847,1281,874,1255,1378,1776,599,1545,410,1005,1514,524,946,701,132,941,750,1566,1306,148,817,503,933,36,906,411,98,1546,1609,820,1625,1584,278,1780,758,310,45,1221,1509,711,349,914,1163,1602,1646,1276,1818,1384,720,1034,1159,1286,1452,291,1489,533,336,12,1640,745,361,1219,1116,721,1717,1019,1138,81,321,592,173,1643,702,496,788,774,1527,952,1345,823,553,333,716,1578,1669,1588,1473,1581,959,1410,465,1470,1234,1352,1742,684,1411,542,1284,944,1425,400,782,915,127,429,1737,1552,378,216,1825,1521,1623,1072,119,995,930,1420,1399,1093,1419,935,1833,1641,807,289,347,1682,1624,158,1379,600,932,586,1651,574,460,1214,676,120,1398,626,1826,1777,394,856,1007,204,473,1156,1679,156,1369,499,591,384,482,1758,927,1105,1793,303,942,1353,1784,41,1456,631,186,960,1551,984,405,304,416,348,47,902,477,1664,707,1706,1530,1830,330,1567,1730,981,74,945,1150,152,113,996,922,849,351,614,488,1429,776,196,596,1711,504,1232,725,425,842,851,435,974,252,677,1691,710,628,88,635,300,699,1193,1715,90,476,769,153,967,700,1298,826,1632,527,1045,18,1820,1638,1343,961,537,1296,1140,343,1596,1512,1741,1134,726,1446,1436,987,1537,1235,356,1303,1587,187,1006,756,283,479,540,653,917,862,575,292,396,1383,1670,589,743,1215,1060,1488,414,96,1634,1476,1677,1123,1827,4,649,1529,824,1675,68,571,1775,1258,852,1155,584,1237,1732,882,1054,1603,843,1197,1099,686,337,1180,218,358,145,658,1391,1787,594,1681,853,1341,1535,185,438,1610,1541,1293,1644,1256,114,1277,1761,642,108,267,1674,399,205,408,1747,1560,1196,1117,825,505,1265,129,1467,236,259,1698,669,573,375,1481,1321,7,670,276,765,140,141,1363,1431,814,94,675,99,1212,1112,1182,61,248,448,815,463,1423,568,1186,483,1090,1131,402,456,1462,1061,1240,1173,1839,1693,247,126,787,1387,1778,393,1201,1359,682,1595,1333,1774,497,485,1842,530,579,993,1463,1822,651,1662,200,1486,338,1585,1364,146,481,58,130,662,44,1358,1109,539,876,1074,1125,325,985,13,749,1209,1290,1736,786,1645,170,1172,436,513,1064,322,962,1477,777,549,1501,1533,620,1709,1591,1043,1687,409,1130,809,50,672,1485,238,1568,731,1309,1037,1821,1113,235,1412,798,1576,491,1791,883,1087,1620,83,1243,1086,353,1160,87,1289,773,1831,1414,198,536,34,535,1164,531,1312) \!\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 $\SL(2,1849)$.

Homology

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

Subgroups

There are 3627279905 subgroups in 123 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 $\PSL(2,1849)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}^2$

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 3160680600 (order at least 1) are shown, as well as all normal subgroups of any index.

Order 3160680600: $\PSL(2,1849)$

Order 1708476: $C_{43}^2:C_{924}$

Order 854238: $C_{43}^2:C_{462}$

Order 569492: $C_{43}^2:C_{308}$

Order 427119: $C_{43}^2:C_{231}$

Order 284746: $C_{43}^2:C_{154}$

Order 244068: $C_{43}^2:C_{132}$

Order 155316: $C_{43}^2:C_{84}$

Order 142373: $C_{43}^2:C_{77}$

Order 122034: $C_{43}^2:C_{66}$

Order 81356: $C_{43}^2:C_{44}$

Order 79464: $\PGL(2,43)$ x 2

Order 77658: $C_{43}^2:C_{42}$

Order 61017: $C_{43}^2:C_{33}$

Order 51772: $C_{43}^2:C_{28}$

Order 40678: $C_{43}^2:C_{22}$

Order 39732: $\PSL(2,43)$ x 2

Order 38829: $C_{43}^2:C_{21}$

Order 25886: $C_{43}^2:C_{14}$

Order 22188: $C_{43}^2:C_{12}$

Order 20339: $C_{43}^2:C_{11}$

Order 12943: $C_{43}^2:C_7$

Order 11094: $C_{43}^2:C_6$

Order 7396: $C_{43}^2:C_4$

Order 5547: $C_{43}^2:C_3$

Order 3698: $C_{43}^2:C_2$

Order 1850: $D_{925}$

Order 1849: $C_{43}^2$

Order 1848: $D_{924}$

Order 1806: $F_{43}$ x 2

Order 925: $C_{925}$

Order 924: $D_{462}$ x 2, $C_{924}$

Order 903: $C_{43}:C_{21}$ x 2

Order 616: $D_{308}$

Order 602: $C_{43}:C_{14}$ x 2

Order 462: $D_{231}$ x 2, $C_{462}$

Order 370: $D_{185}$

Order 308: $D_{154}$ x 2, $C_{308}$

Order 301: $C_{43}:C_7$ x 2

Order 264: $D_{132}$

Order 258: $C_{43}:C_6$ x 2

Order 231: $C_{231}$

Order 185: $C_{185}$

Order 168: $D_{84}$

Order 154: $D_{77}$ x 2, $C_{154}$

Order 132: $D_{66}$ x 2, $C_{132}$

Order 129: $C_{43}:C_3$ x 2

Order 88: $D_{44}$

Order 86: $D_{43}$ x 2

Order 84: $D_{42}$ x 2, $C_{84}$

Order 77: $C_{77}$

Order 74: $D_{37}$

Order 66: $D_{33}$ x 2, $C_{66}$

Order 60: $A_5$ x 2

Order 56: $D_{28}$

Order 50: $D_{25}$

Order 44: $D_{22}$ x 2, $C_{44}$

Order 43: $C_{43}$ x 2

Order 42: $D_{21}$ x 2, $C_{42}$

Order 37: $C_{37}$

Order 33: $C_{33}$

Order 28: $D_{14}$ x 2, $C_{28}$

Order 25: $C_{25}$

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

Order 22: $D_{11}$ x 2, $C_{22}$

Order 21: $C_{21}$

Order 14: $D_7$ x 2, $C_{14}$

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

Order 11: $C_{11}$

Order 10: $D_5$

Order 8: $D_4$

Order 7: $C_7$

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 3160680600: $\PSL(2,1849)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,1849)$)

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 $927 \times 927$ character table is not available for this group.

Rational character table

The $31 \times 31$ rational character table is not available for this group.