Group #72221415

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	S₄×GL(3,𝔽₂)
commutator, hypocenter	#72209751	C₂
Fitting	Q₈	D₃×GL(3,𝔽₂)
cosocle	SL(2,𝔽₃)	C₂×GL(3,𝔽₂)
LCS	G ▷ #72209751
UCS	C₁ ◁ C₂

subgroup

G/H

center, Frattini, socle, hypercenter

C₂

S₄×GL(3,𝔽₂)

commutator, hypocenter

#72209751

C₂

Fitting

Q₈

D₃×GL(3,𝔽₂)

C₂×GL(3,𝔽₂)

G ▷ #72209751

C₁ ◁ C₂

347 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Presentation

Permutation generators:

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)(1037,1341,1420,1459)(1038,1347,1419,1453)(1039,1353,1418,1447)(1040,1359,1417,1441)(1041,1365,1416,1428)(1042,1329,1415,1422)(1043,1340,1407,1460)(1044,1346,1413,1454)(1045,1352,1412,1448)(1046,1358,1411,1435)(1047,1371,1410,1429)(1048,1328,1409,1423)(1049,1334,1408,1466)(1050,1345,1400,1455)(1051,1351,1406,1442)(1052,1364,1405,1436)(1053,1370,1404,1430)(1054,1327,1403,1424)(1055,1333,1402,1467)(1056,1339,1401,1461)(1057,1357,1393,1443)(1058,1363,1399,1437)(1059,1369,1398,1431)(1060,1326,1397,1425)(1061,1332,1396,1468)(1062,1338,1395,1462)(1063,1344,1394,1449)(1064,1362,1386,1438)(1065,1368,1392,1432)(1066,1325,1391,1426)(1067,1331,1390,1469)(1068,1337,1389,1456)(1069,1350,1388,1450)(1070,1356,1387,1444)(1071,1367,1379,1433)(1072,1324,1385,1427)(1073,1330,1384,1463)(1074,1343,1383,1457)(1075,1349,1382,1451)(1076,1355,1381,1445)(1077,1361,1380,1439)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,295,6,55)(2,247,5,103)(3,199,4,151)(7,350,42,2100)(8,645,48,2155)(9,597,47,2203)(10,549,46,2251)(11,501,45,2299)(12,453,44,2347)(13,405,43,2395)(14,700,35,1750)(15,995,41,1805)(16,947,40,1853)(17,899,39,1901)(18,851,38,1949)(19,803,37,1997)(20,755,36,2045)(21,1050,28,1400)(22,1345,34,1455)(23,1297,33,1503)(24,1249,32,1551)(25,1201,31,1599)(26,1153,30,1647)(27,1105,29,1695)(49,296,294,54)(50,248,300,102)(51,200,299,150)(52,152,298,198)(53,104,297,246)(56,646,336,2154)(57,598,342,2202)(58,550,341,2250)(59,502,340,2298)(60,454,339,2346)(61,399,338,2394)(62,351,337,2106)(63,996,329,1804)(64,948,335,1852)(65,900,334,1900)(66,852,333,1948)(67,804,332,1996)(68,749,331,2044)(69,701,330,1756)(70,1346,322,1454)(71,1298,328,1502)(72,1250,327,1550)(73,1202,326,1598)(74,1154,325,1646)(75,1099,324,1694)(76,1051,323,1406)(77,1696,315,1104)(78,1648,321,1152)(79,1600,320,1200)(80,1552,319,1248)(81,1504,318,1296)(82,1449,317,1344)(83,1401,316,1056)(84,2046,308,754)(85,1998,314,802)(86,1950,313,850)(87,1902,312,898)(88,1854,311,946)(89,1799,310,994)(90,1751,309,706)(91,2396,301,404)(92,2348,307,452)(93,2300,306,500)(94,2252,305,548)(95,2204,304,596)(96,2149,303,644)(97,2101,302,356)(98,249,245,101)(99,201,251,149)(100,153,250,197)(105,599,287,2201)(106,551,293,2249)(107,503,292,2297)(108,448,291,2345)(109,400,290,2400)(110,352,289,2105)(111,647,288,2153)(112,949,280,1851)(113,901,286,1899)(114,853,285,1947)(115,798,284,1995)(116,750,283,2050)(117,702,282,1755)(118,997,281,1803)(119,1299,273,1501)(120,1251,279,1549)(121,1203,278,1597)(122,1148,277,1645)(123,1100,276,1700)(124,1052,275,1405)(125,1347,274,1453)(126,1649,266,1151)(127,1601,272,1199)(128,1553,271,1247)(129,1498,270,1295)(130,1450,269,1350)(131,1402,268,1055)(132,1697,267,1103)(133,1999,259,801)(134,1951,265,849)(135,1903,264,897)(136,1848,263,945)(137,1800,262,1000)(138,1752,261,705)(139,2047,260,753)(140,2349,252,451)(141,2301,258,499)(142,2253,257,547)(143,2198,256,595)(144,2150,255,650)(145,2102,254,355)(146,2397,253,403)(147,202,196,148)(154,552,238,2248)(155,497,244,2296)(156,449,243,2351)(157,401,242,2399)(158,353,241,2104)(159,648,240,2152)(160,600,239,2200)(161,902,231,1898)(162,847,237,1946)(163,799,236,2001)(164,751,235,2049)(165,703,234,1754)(166,998,233,1802)(167,950,232,1850)(168,1252,224,1548)(169,1197,230,1596)(170,1149,229,1651)(171,1101,228,1699)(172,1053,227,1404)(173,1348,226,1452)(174,1300,225,1500)(175,1602,217,1198)(176,1547,223,1246)(177,1499,222,1301)(178,1451,221,1349)(179,1403,220,1054)(180,1698,219,1102)(181,1650,218,1150)(182,1952,210,848)(183,1897,216,896)(184,1849,215,951)(185,1801,214,999)(186,1753,213,704)(187,2048,212,752)(188,2000,211,800)(189,2302,203,498)(190,2247,209,546)(191,2199,208,601)(192,2151,207,649)(193,2103,206,354)(194,2398,205,402)(195,2350,204,450)(343,2093,2058,357)(344,2388,2064,412)(345,2340,2063,460)(346,2292,2062,508)(347,2244,2061,556)(348,2196,2060,604)(349,2148,2059,652)(358,638,2099,2113)(359,590,2098,2161)(360,542,2097,2209)(361,494,2096,2257)(362,446,2095,2305)(363,398,2094,2353)(364,693,2086,1757)(365,988,2092,1812)(366,940,2091,1860)(367,892,2090,1908)(368,844,2089,1956)(369,796,2088,2004)(370,748,2087,2052)(371,1043,2079,1407)(372,1338,2085,1462)(373,1290,2084,1510)(374,1242,2083,1558)(375,1194,2082,1606)(376,1146,2081,1654)(377,1098,2080,1702)(378,1393,2072,1057)(379,1688,2078,1112)(380,1640,2077,1160)(381,1592,2076,1208)(382,1544,2075,1256)(383,1496,2074,1304)(384,1448,2073,1352)(385,1743,2065,707)(386,2038,2071,762)(387,1990,2070,810)(388,1942,2069,858)(389,1894,2068,906)(390,1846,2067,954)(391,1798,2066,1002)(392,2389,2352,411)(393,2341,2358,459)(394,2293,2357,507)(395,2245,2356,555)(396,2197,2355,603)(397,2142,2354,651)(406,639,2387,2112)(407,591,2393,2160)(408,543,2392,2208)(409,495,2391,2256)(410,447,2390,2304)(413,989,2380,1811)(414,941,2386,1859)(415,893,2385,1907)(416,845,2384,1955)(417,797,2383,2003)(418,742,2382,2051)(419,694,2381,1763)(420,1339,2373,1461)(421,1291,2379,1509)(422,1243,2378,1557)(423,1195,2377,1605)(424,1147,2376,1653)(425,1092,2375,1701)(426,1044,2374,1413)(427,1689,2366,1111)(428,1641,2372,1159)(429,1593,2371,1207)(430,1545,2370,1255)(431,1497,2369,1303)(432,1442,2368,1351)(433,1394,2367,1063)(434,2039,2359,761)(435,1991,2365,809)(436,1943,2364,857)(437,1895,2363,905)(438,1847,2362,953)(439,1792,2361,1001)(440,1744,2360,713)(441,2342,2303,458)(442,2294,2309,506)(443,2246,2308,554)(444,2191,2307,602)(445,2143,2306,657)(455,592,2338,2159)(456,544,2344,2207)(457,496,2343,2255)(461,640,2339,2111)(462,942,2331,1858)(463,894,2337,1906)(464,846,2336,1954)(465,791,2335,2002)(466,743,2334,2057)(467,695,2333,1762)(468,990,2332,1810)(469,1292,2324,1508)(470,1244,2330,1556)(471,1196,2329,1604)(472,1141,2328,1652)(473,1093,2327,1707)(474,1045,2326,1412)(475,1340,2325,1460)(476,1642,2317,1158)(477,1594,2323,1206)(478,1546,2322,1254)(479,1491,2321,1302)(480,1443,2320,1357)(481,1395,2319,1062)(482,1690,2318,1110)(483,1992,2310,808)(484,1944,2316,856)(485,1896,2315,904)(486,1841,2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order	8064
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

#72221415

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP