Abstract group 6920642400.a: $\PSL(2,2401)$

• Label: 6920642400.a
• Order: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{4} \cdot 1201 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 1201 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5^{2} \cdot 7^{4} \cdot 1201 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $2402$
• Trans deg.: $2402$
• Rank: $2$

Group information

Description:	$\PSL(2,2401)$
Order:	\(6920642400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{4} \cdot 1201 \)
Exponent:	\(10088400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 1201 \)
Automorphism group:	Group of order \(55365139200\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \cdot 7^{4} \cdot 1201 \)
Composition factors:	$\PSL(2,2401)$
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	8	10	12	15	16	20	24	25	30	40	48	50	60	75	80	100	120	150	200	240	300	400	600	1200	1201
Elements	1	2883601	5767202	5767202	11534404	5767202	5764800	11534404	11534404	11534404	23068808	23068808	23068808	23068808	57672020	23068808	46137616	46137616	57672020	46137616	115344040	92275232	115344040	92275232	115344040	230688080	184550464	230688080	461376160	461376160	922752320	3457440000	6920642400
Conjugacy classes	1	1	1	1	2	1	2	2	2	2	4	4	4	4	10	4	8	8	10	8	20	16	20	16	20	40	32	40	80	80	160	600	1203
Divisions	1	1	1	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	33
Autjugacy classes	1	1	1	1	1	1	1	2	1	1	1	2	1	2	5	1	2	4	5	2	5	4	5	4	5	10	8	10	20	20	40	150	317

Minimal presentations

Permutation degree:	$2402$
Transitive degree:	$2402$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1201	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\PSL(2,2401)$, $\OmegaMinus(4,49)$, $\POmegaMinus(4,49)$
Permutation group:	Degree $2402$ $\langle(3,161,174,1235,1111,837,399,641,334,2227,1911,1794,69,169,1666,411,60,1844,1542,2392,1974,506,2056,327,1671,1432,1509,565,392,144,651,1580,879,592,282,1296,1016,586,328,1491,869,92,1562,409,2374,1765,515,1456,1133,1775,1535,50,1691,2059,249,2195,819,395,814,1108,296,1860,719,492,2191,1829,1090,906,192,536,1127,1334,1712,2361,1966,190,652,776,1015,2012,2100,2239,426,2189,1892,2342,633,775,1977,432,994,1821,1855,1139,2362,1010,168,1872,2174,1323,624,1072,312,1496,287,244,1337,956,403,1548,34,551,1029,15,1371,497,1818,1228,1564,2124,1126,998,1441,373,1903,1782,2211,1205,170,1384,965,90,358,2277,554,2164,1832,1353,1252,1058,2183,808,1098,1428,191,564,1179,958,891,961,885,1724,140,1442,2166,2043,1244,130,1698,1167,171,893,1085,2219,2038,471,778,1993,2271,1110,375,457,1610,567,1368,772,934,1511,250,286,1103,2041,597,2354,1150,1092,424,943,1005,1264,2396,605,2212,2199,803,1067,785,1124,1050,797,1237,154,1874,716,582,580,2352,566,1263,2217,1118,2376,1242,556,936,2240,676,259,588,671,1403,2003,822,1768,1959,1211,1789,1795,1717,2316,1081,1581,815,1957,135,1057,1990,1257,1197,2196,1796,1559,1517,478,57,1426,80,1012,984,1617,828,2334,1319,2070,1345,748,1878,1524,2112,1658,1024,339,529,590,540,985,1474,461,311,2363,262,394,2042,680,1939,1195,834,1599,1672,1299,213,245,113,1773,1700,1703,1201,1398,1379,886,418,966,2082,145,2168,390,1739,1365,655,833,890,1013,1914,2347,686,476,1077,1778,1493,1762,2360,905,1069,133,1038,631,2322,76,279,594,876,348,408,2190,696,423,1545,217,1128,1030,2300,324,36,2203,226,900,40,2130,2208,378,1945,1036,1054,2172,1770,1556,414,813,1132,1413,871,2167,970,741,743,340,1898,1679,1730,203,101,1495,1713,1946,1907,1344,2128,1569,1488,1254,1102,1461,1502,807,1401,1924,391,1117,1422,653,2229,1003,496,2062,581,1056,179,1506,1035,2053,2302,1091,434,480,290,866,102,377,1859,1523,1203,663,842,1358,1247,2305,2106,1537,2142,511,947,175,1688,67,205,2343,427,673,1504,1095,1478,1311,2105,150,333,1983,1451,1964,2274,39,280,55,1528,1011,2254,1280,2016,1651,317,1224,2111,1032,520,1450,1801,26,525,1663,1346,2193,974,157,1149,2337,1064,1073,1943,940,1089,1646,1791,694,153,1980,48,379,1116,2256,589,376,2052,1607,544,454,770,1578,155,1538,1565,538,1871,2268,945,366,435,183,981,298,800,1685,593,487,2262,104,2173,2122,1174,1343,691,2243,2206,1906,87,463,678,1737,1632,1707,266,1159,957,2220,650,1501,2131,2185,2378,1261,1383,1269,1743,816,2266,2186,182,2216,1895,1105,148,2037,600,505,731,1798,1234,111,2326,851,19,1941,2221,1433,452,2050,1988,1575,59,608,517,1560,337,976,2249,1905,1473,1668,2388,991,855,35,1184,853,1275,387,1173,697,1366,978,942,811,596,968,2285,1399,543,1468,136,1178,518,28,1751,1596,1616,295,404,875,1410,121,923,11,2317,1676,558,1327,1446,2204,768,1483,443,1454,309,1624,383,730,1217,1129,1361,1207,147,2099,672,2246,307,1531,658,1028,1856,1466,21,2307,2127,1555,1896,223,1119,784,357,1025,1378,524,780,292,261,1472,591,887,1916,164,1613,1156,1018,537,1793,473,1828,1682,122,23,163,372,1601,2397,1567,996,727,265,1992,622,1661,2021,158,1639,2103,109,46,1786,75,1748,112,646,484,2102,2327,1835,2040,1521,1049,318,1325,1443,138,1716,351,107,2297,386,2335,547,25,1967,1845,284,388,486,684,118,1780,501,713,1065,95,469,1619,1785,1792,1807,1258,1063,1053,405,762,1508,1803,115,1694,512,835,1741,721,2074,2104,1744,1364,1701,777,482,483,2398,526,1391,1009,1492,634,18,2373,583,2320,1735,1284,30,5,126,832,569,2383,252,854,1239,1350,1937,1634,2351,313,699,1579,1799,1593,1866,2046,1046,1837,1655,1154,208,760,703,1935,2085,110,1547,892,514,242,1362,2364,200,224,578,453,1963,685,2023,1870,1965,271,1623,1453,116,647,2045,574,665,456,449,1718,738,599,758,84,332,2097,1486,921,1475,1245,1749,791,1857,2324,371,621,1659,861,1889,638,1076,2061,1455,2120,1182,532,2133,354,479,1335,2233,2201,1728,1704,898,1143,1527,682,1209,180,530,986,1497,420,523,724,1863,1317,1853,1271,61,1920,1412,2176,1213,1516,769,1097,2282,1306,2129,579,1687,975,1385,1301,1833,562,559,472,839,1000,2049,967,1553,1408,258,865,508,1477,2311,2075,1746,1804,24,477,2218,1418,2063,709,2180,615,2076,2116,1485,1292,416,1225,117,817,1787,436,553,911,1176,66,1033,988,1096,1660,2280,1766,990,1250,916,1100,326,2323,1157,1374,629,314,1048,872,196,859,1584,989,626,926,2069,347,467,397,264,1915,1331,310,571,1404,2368,381,91,2328,1318,1470,1638,2175,820,240,1189,2393,519,1677,700,173,1582,1397,2101,1114,198,763,2187,2263,1162,475,818,451,1330,211,1975,1172,649,1293,563,269,2319,1763,421,1282,805,207,97,1083,2108,751,1961,1715,801,1574,437,195,1218,1736,322,407,555,350,181,1680,695,2150,847,864,493,1169,1152,1940,895,2004,234,1745,52,873,124,528,848,953,2044,1902,2090,1017,447,141,749,2341,915,683,617,2036,1481,2401,752,6,1947,1120,1290,2331,1769,419,2245,1140,1802,119,325,2117,257,1019,1665,1690,753,10,2312,2137,167,1165,106,2138,396,374,2132,1183,370,291,1649,1215,1283,735,1268,2157,1820,56,474,1699,1693,1848,1457,127,1360,1382,938,20,1611,2259,1445,1777,221,1697,1191,787,2159,929,902,1206,810,1522,1673,2291,2025,1615,1326,2064,2158,270,1810,1622,1107,2333,228,1576,1212,2086,47,99,2389,2276,1901,1683,2367,1648,1738,1631,1135,445,918,433,1950,2356,761,1259,1241,1525,802,2084,1626,575,2391,862,1084,1260,1549,1219,1883,1339,1650,410,1185,235,398,674,644,1459,1726,1175,799,2340,1406,2151,541,260,1131,251,1976,1917,1423,1757,1910,1640,132,1695,33,2020,907,1609,1093,236,1138,2005,243,1354,925,2089,227,1304,2152,1760,338,177,980,1641,1669,941,406,2149,2060,969,2309,2332,793,1561,1955,1363,1884,971,152) \!\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 $\SL(2,2401)$.

Homology

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

Subgroups

There are 2 normal subgroups, 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 $\PSL(2,2401)$
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:	$P_{ 5 } \simeq$ $C_{25}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4$
1201-Sylow subgroup:	$P_{ 1201 } \simeq$ $C_{1201}$

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 6591088 (order at least 1050) are shown, as well as all normal subgroups of any index.

Order 6920642400: $\PSL(2,2401)$

Order 2881200: $C_7^4.C_{300}.C_4$

Order 1440600: $C_7^4.C_{75}.C_8$

Order 960400: $C_7^4.C_{400}$

Order 720300: $C_7^4.C_{300}$

Order 576240: $C_7^4.C_{240}$

Order 480200: $C_7^4.C_{200}$

Order 360150: $C_7^4.C_{150}$

Order 288120: $C_7^4.C_{120}$

Order 240100: $C_7^4.C_{100}$

Order 192080: $C_7^4.C_{80}$

Order 180075: $C_7^4:C_{75}$

Order 144060: $C_7^4.C_{60}$

Order 120050: $C_7^4:C_{50}$

Order 117600: $\PGL(2,49)$ x 2

Order 115248: $C_7^4.C_{48}$

Order 96040: $C_7^4.C_{40}$

Order 72030: $C_7^4:C_{30}$

Order 60025: $C_7^4:C_{25}$

Order 58800: $\PSL(2,49)$ x 2

Order 57624: $C_7^4.C_{24}$

Order 48020: $C_7^4:C_{20}$

Order 38416: $C_7^4.C_{16}$

Order 36015: $C_7^4:C_{15}$

Order 28812: $C_7^4:C_{12}$

Order 24010: $C_7^4:C_{10}$

Order 19208: $C_7^4:C_8$

Order 14406: $C_7^4:C_6$

Order 12005: $C_7^4:C_5$

Order 9604: $C_7^4:C_4$

Order 7203: $C_7^4:C_3$

Order 4802: $C_7^4:C_2$

Order 2402: $D_{1201}$

Order 2401: $C_7^4$

Order 2400: $D_{1200}$

Order 2352: $F_{49}$ x 2

Order 2058: $C_7^3:C_6$ x 2

Order 1201: $C_{1201}$

Order 1200: $D_{600}$ x 2, $C_{1200}$

Order 1176: $C_7^2:C_{24}$ x 2

Order 32: $D_{16}$

Order 25: $C_{25}$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 6920642400: $\PSL(2,2401)$ ($C_1$)

Order 1: $C_1$ ($\PSL(2,2401)$)

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 $1203 \times 1203$ character table is not available for this group.

Rational character table

The $33 \times 33$ rational character table is not available for this group.