LMFDB - Abstract group 1442784.a: $\SL(2,113)$

Show commands: Gap / Magma / SageMath

magma:G := SL(2, 113);

gap:G := SL(2, 113);

sage:G = SL(2, 113)

comment:Define the group as a permutation group

Group information

Description:	$\SL(2,113)$
Order:	$1442784$$\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 19 \cdot 113 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$721392$$\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 19 \cdot 113 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$\PGL(2,113)$, of order $1442784$$\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 19 \cdot 113 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$, $\PSL(2,113)$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$0$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and quasisimple (hence nonsolvable and perfect). Whether it is almost simple has not been computed.

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

Order	1	2	3	4	6	7	8	14	16	19	28	38	56	57	112	113	114	226
Elements	1	1	12656	12882	12656	38646	25764	38646	51528	113904	77292	113904	154584	227808	309168	12768	227808	12768	1442784
Conjugacy classes	1	1	1	1	1	3	2	3	4	9	6	9	12	18	24	2	18	2	117
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	18
Autjugacy classes	1	1	1	1	1	3	2	3	4	9	6	9	12	18	24	1	18	1	115

Minimal presentations

Permutation degree:	$1824$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	56	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Show commands: Gap / Magma / SageMath

Groups of Lie type:	$\SL(2,113)$, $\SU(2,113)$
Permutation group:	Degree $1824$ $\langle(1,2,6,24,149,810,1764,1203,1352,1761,1821,1631,563,95,16,4)(3,9,39,248,1152,1734,1075,462,624,1037,1658,1753,790,1559,922,172,748,1225,540,1547,1392,1428,862,260,801,580,1053,1708,716,1241,345,52,10,48,316,1332,1767,1736,760,138,22,134,749,1015,1129,1544,686,619,130,739,1192,1469,1524,450,1183,735,745,608,1132,1651,1755,1704,1403,970,327,93,556,1625,1785,870,160,868,752,456,570,1646,1792,1606,1557,584,146,798,1545,470,362,956,964,232,132,346,847,1615,1266,290,832,435,1209,1201,1165,1822,1396,452,864,1093,332,577,1641,643,1130,445,69,12) \!\cdots\! \rangle$ 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comment:Define the group as a permutation group magma:G := PermutationGroup< 1824 \| 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>; gap:G := Group( 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); sage:G = PermutationGroup(['(1,2,6,24,149,810,1764,1203,1352,1761,1821,1631,563,95,16,4)(3,9,39,248,1152,1734,1075,462,624,1037,1658,1753,790,1559,922,172,748,1225,540,1547,1392,1428,862,260,801,580,1053,1708,716,1241,345,52,10,48,316,1332,1767,1736,760,138,22,134,749,1015,1129,1544,686,619,130,739,1192,1469,1524,450,1183,735,745,608,1132,1651,1755,1704,1403,970,327,93,556,1625,1785,870,160,868,752,456,570,1646,1792,1606,1557,584,146,798,1545,470,362,956,964,232,132,346,847,1615,1266,290,832,435,1209,1201,1165,1822,1396,452,864,1093,332,577,1641,643,1130,445,69,12)(5,17,96,564,1632,1809,1556,925,330,265,1202,1766,812,151,26,8)(7,27,164,890,1790,1467,1474,413,1042,634,661,1680,1140,1536,1279,830,621,266,41,261,1187,1323,701,897,666,112,609,1581,1098,786,933,175,28,171,314,541,1489,1528,453,238,37,235,1120,1303,1788,1808,1080,444,231,519,549,1620,1542,991,1555,1111,802,147,800,623,1689,1594,1258,527,85,14,81,506,1296,1663,820,1181,257,136,102,606,1570,1648,688,117,245,534,268,1001,942,1509,1153,518,233,121,705,1591,1633,910,1339,981,405,322,280,1237,1418,993,1357,1497,650,109,589,1055,1696,807,188,30)(11,56,368,1417,475,1553,969,223,639,502,1135,1706,1731,1121,958,1305,304,46,300,1295,1299,516,82,512,694,1375,1811,1013,576,778,924,180,334,548,1334,1297,1510,965,183,33,205,1038,1479,1634,1578,733,433,67,83,295,1284,1709,1367,903,249,237,720,1576,1824,1424,766,139,763,1023,520,1248,1412,1395,353,1346,954,691,140,770,1551,1573,1257,287,855,928,326,299,1293,495,1438,1104,228,226,133,616,1337,1710,784,757,1373,1034,312,47,309,1316,1294,1354,1018,759,653,1659,1450,1446,1439,1270,401,61)(13,73,460,466,1380,374,1125,738,667,660,1530,656,683,960,182,29,79,494,1161,1003,1443,379,869,684,141,772,1685,1715,1269,1700,1492,918,170,916,1586,1463,1028,198,776,1087,669,1653,644,108,640,1702,835,356,54,283,1163,1413,402,838,158,857,805,1371,568,671,1109,979,186,199,914,1601,1230,1723,1422,891,881,734,712,1656,1342,638,239,1126,689,1031,395,1228,1180,938,1386,451,1084,240,633,421,1484,1712,907,1245,439,68,436,1285,1022,1608,1786,873,730,234,746,535,1486,1137,1123,1327,478,76)(15,90,472,1415,1692,1437,681,1320,422,537,827,1776,1246,1348,324,49,272,1222,1529,1044,1783,860,211,680,267,1211,641,1703,1243,1122,441,333,91,546,1587,1301,819,1142,762,602,101,599,243,488,768,1252,529,499,595,1419,1716,1802,1102,1124,1070,1672,1205,1208,773,1372,988,1004,828,419,1198,561,1148,1800,1097,224,35,220,1088,383,1448,1032,1255,957,1240,281,43,277,949,179,946,397,407,131,597,1012,194,1008,1743,929,174,339,1206,1456,1714,1816,1514,874,216,704,1216,1447,1499,1408,769,620,104,75)(18,103,610,1682,1307,1520,1100,1464,474,463,1233,1388,1803,976,274,256,137,754,1483,1812,1813,1039,603,1351,496,275,1162,1157,1763,1433,1105,394,1065,604,1675,1737,1756,1605,648,203,337,1174,254,983,369,1421,727,127,125,598,1649,1011,1261,1679,1007,1020,526,1546,545,1619,1398,1173,951,491,649,1655,1820,1599,1282,1741,1024,197,554,92,550,1621,947,449,1478,418,122,710,1330,780,911,1733,750,399,1452,412,544,1617,1394,1691,1229,557,721,699,1725,1575,486,987,996,329,59,386,902,936,1797,1191,408,62)(19,111,658,1609,1823,1740,953,181,89,539,1218,1819,1382,1440,871,206,113,668,1056,1356,811,1560,1503,921,799,893,543,1616,1807,1026,1207,1107,771,605,567,1639,1744,1081,1720,943,415,479,980,1804,1384,1523,889,1086,1136,1664,1344,1360,336,1358,848,826,1353,510,833,843,1768,1513,513,1133,629,593,473,1432,1309,1256,834,528,1312,306,1311,1724,1400,507,1227,411,672,614,1686,1697,1752,817,190,184,533,273,1226,1156,301,824,163,885,679,575,1635,1600,791,1754,962,430,915,1238,1333,1399,354,1029,682,116)(20,119,695,509,1577,523,934,398,1461,1199,542,1436,377,57,373,1073,1066,596,1569,485,77,481,1366,340,51,335,1355,1588,899,1719,1108,229,115,677,1534,1283,913,1274,446,1025,572,311,94,454,1324,1442,1471,1092,804,1490,1552,908,168,905,974,952,363,547,999,1543,1423,628,1050,208,110,423,1488,1698,1596,1727,978,755,360,508,559,937,1387,865,159,793,706,1644,569,788,144,765,856,926,900,372,250,1158,1369,1472,723,758,670,1507,1611,1593,504,1517,443,477,169,912,1166,1630,803,1434,715,123)(21,126,722,1687,1654,1770,1236,622,904,1179,255,1176,1435,1579,1064,410,118,690,1482,1815,1750,1242,892,525,385,279,1168,1583,1411,1726,1453,714,709,782,1747,1139,1729,841,1350,325,492,1214,1217,1799,1370,1773,1470,1134,665,574,1650,1014,1500,846,156,72,50,328,469,74,465,1538,1186,259,626,587,1666,1571,1263,289,849,344,741,1304,1806,1155,1368,1194,884,632,625,726,1647,1699,1127,1271,1224,531,498,600,1278,1381,349,1378,724,796,145,792,1637,1795,1793,1281,530,86,87,302,1169,1154,1707,1049,736,129)(23,143,779,615,1082,218,887,428,500,308,1189,1604,1416,1030,877,222,794,1638,566,1574,744,1197,963,400,536,553,1585,1624,1364,1511,1059,210,124,573,1636,1150,842,155,25,152,740,298,1171,1213,366,458,1531,1048,743,785,1341,1475,1397,1521,447,348,166,898,742,471,1091,1493,1067,213,148,806,1676,646,483,286,153,490,78,42,270,351,53,70,162,879,1701,751,1614,1668,1674,767,876,365,55,361,1175,1468,1300,1441,1147,246,1146,697,1678,1519,717,737,201,32,58,307,251,674,341,440,258,40)(31,63,269,998,1322,310,378,1114,1068,753,651,1061,1077,1565,837,154,196,1021,1235,1231,1506,950,1758,1078,241,1131,1652,1671,1794,1718,764,1512,829,1250,284,1190,1782,850,713,1495,1069,630,607,209,1052,1310,1262,940,177,338,558,1535,961,1264,818,635,1149,594,100,591,1669,1592,840,851,1349,1210,276,1170,355,1393,1526,1110,659,654,585,1051,882,1717,1083,1167,252,392,511,1602,1798,992,1784,883,1047,678,1212,1677,1751,789,985,187,982,524,264,1200,1541,1410,1106,878,1118,775,613,1612,1580,487,1005,193)(34,212,1063,1713,1460,1244,282,836,204,331,263,1164,1550,1539,1693,858,214,655,1661,1345,1765,1562,1331,1409,618,296,45,292,1275,1781,1377,1473,777,142,99,586,1598,1626,1239,1251,347,1006,1314,1277,1459,1554,1622,551,728,675,663,1391,927,1325,313,357,1402,384,317,1336,1711,1195,262,1193,732,128,65,420,1481,1280,293,1035,404,505,1223,1319,562,381,1429,968,1071,702,1681,1814,1260,288,44,285,427,389,320,1338,468,1302,823,1427,637,107,578,1451,1141,1306,1313,977,431,1476,1328,1420,939,352,1076,217)(36,227,1101,1670,647,1259,1273,448,1494,532,894,1318,1548,1759,797,967,219,664,1657,1690,1749,787,1728,1033,457,342,1249,1618,1389,1379,376,1096,1094,711,1491,1613,1818,1566,1406,923,1019,521,464,1232,1390,1745,1757,1694,631,106,617,1688,1518,1769,816,191,173,359,1000,192,997,1597,1445,896,685,120,700,1449,1405,909,1365,932,438,1363,1425,1178,1796,1791,1775,1046,1043,692,612,1684,1787,1732,747,424,426,135,294,1177,935,1665,1103,1144,244,627,582,1662,1308,305,1036,202,66,84,522,1335,1505,1466,1112,230)(38,242,343,1041,1735,866,1247,975,403,920,297,1288,1221,1762,1515,517,406,583,98,579,1117,1730,1778,959,393,60,390,388,1455,1738,1430,861,225,105,581,1660,645,815,150,813,1115,323,278,253,945,995,1516,514,1116,731,1487,1533,1572,1359,989,687,825,364,319,1002,434,1504,1196,863,247,693,781,1057,1010,906,814,1017,195,167,901,497,176,189,822,1774,1627,774,698,719,761,1128,1772,944,178,367,1234,1287,1501,1540,1789,888,1584,1090,783,808,1099,1113,853,157,71,64,416,1072,930,986,895,165)(80,501,560,1628,1431,375,966,1062,1315,588,1465,1810,1272,291,1268,476,917,1290,461,1185,1801,1760,1182,1060,707,1343,321,1340,1748,1326,442,459,1215,552,1159,271,350,1383,859,1151,555,97,571,1009,1276,1253,1502,718,1407,703,417,1477,1817,1496,429,1143,708,601,1498,1265,1582,1683,611,955,1347,370,391,1457,1188,1027,1184,729,756,662,1645,1522,1779,1742,880,455,88,371,1089,467,437,1508,1074,215,592,590,1643,1568,484,839,852,480,493,1204,1172,1016,1771,821,1454,387,636,1610,1054,1746,1780,1561,972,503)(114,673,1642,919,1361,854,185,200,358,1291,1805,1219,489,875,161,872,1623,725,1058,642,1705,990,1404,425,538,315,1329,1629,1145,1532,867,409,657,1485,1640,1138,1567,1739,994,931,1286,303,1298,1317,948,1444,1603,1095,652,1374,1537,1607,1595,1722,1267,432,414,1289,1589,1695,1401,1527,1045,207,809,1590,565,1563,482,844,831,941,1362,845,1667,1220,1414,1721,1079,221,795,1376,1558,1777,1564,1254,984,1462,396,1292,1458,1549,1385,1525,515,1085,1040,1673,1426,1480,1119,971,973,236,382,318,1160,696,1321,380,886,676)', 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Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{113})$
comment:Define the group as a matrix group with coefficients in GLFp magma:G := MatrixGroup< 2, GF(113) \| [[1, 1, 0, 1], [1, 0, 1, 1]] >; gap:G := Group([[[ Z(113)^0, Z(113)^0 ], [ 0Z(113), Z(113)^0 ]], [[ Z(113)^0, 0Z(113) ], [ Z(113)^0, Z(113)^0 ]]]); sage:MS = MatrixSpace(GF(113), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[1, 0], [1, 1]])])
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SL(2,113)$.

Homology

Abelianization:	$C_1 $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	not computed	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

Subgroup data has not been computed.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

The $117 \times 117$ character table is not available for this group.

Rational character table

The $18 \times 18$ rational character table is not available for this group.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	1442784.a
Order	\( 2^{5} \cdot 3 \cdot 7 \cdot 19 \cdot 113 \)
Exponent	\( 2^{4} \cdot 3 \cdot 7 \cdot 19 \cdot 113 \)

Nilpotent	no
Solvable	no
$\card{G^{\mathrm{ab}}}$	\( 1 \)
$\card{Z(G)}$	2
$\card{\Aut(G)}$	\( 2^{5} \cdot 3 \cdot 7 \cdot 19 \cdot 113 \)
$\card{\mathrm{Out}(G)}$	\( 2 \)
Perm deg.	$1824$
Trans deg.	not computed
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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