Finite Group Databasebeta

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C₄₅₆₁⋊C₅ #37503590

• metacyclic
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem
• Frobenius

Aliases

Splittings

Subgroups

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 4561 points:

(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,105)(10,19,36,18,34)(12,23,44,82,90)(14,27,52,97,64)(16,31,60,113,204)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,155,189)(26,50,93,35,66)(28,54,101,181,163)(30,58,109,196,121)(32,62,117,212,366)(37,69,41,76,68)(39,73,133,106,190)(43,80,144,148,164)(45,84,151,174,122)(47,88,159,275,351)(51,95,170,92,165)(53,99,177,137,123)(55,103,185,317,325)(57,107,192,67,124)(59,111,200,343,283)(61,115,208,358,220)(63,119,216,374,524)(70,128,83,149,167)(72,131,98,175,126)(74,135,238,205,352)(77,139,130,94,168)(79,142,141,138,127)(81,146,253,268,326)(85,153,264,310,284)(87,157,271,336,221)(89,161,279,462,601)(96,172,295,191,327)(100,179,306,257,285)(102,183,313,299,222)(104,187,321,525,575)(108,194,332,166,286)(110,198,339,242,223)(112,202,347,567,533)(114,206,354,125,224)(116,210,362,593,470)(118,214,370,608,381)(120,218,378,623,978)(129,229,156,269,329)(132,233,182,311,288)(134,236,197,337,226)(136,240,406,367,602)(140,245,235,193,330)(143,248,250,258,289)(145,251,261,300,227)(147,255,425,455,576)(150,259,292,169,290)(152,262,303,243,228)(154,266,440,518,534)(158,273,451,560,471)(160,277,458,586,382)(162,281,466,740,819)(171,293,176,301,231)(173,297,488,353,577)(178,304,247,244,232)(180,308,503,444,535)(184,315,514,507,472)(186,319,521,549,383)(188,323,529,827,940)(195,334,545,328,536)(199,341,556,429,473)(201,345,563,492,384)(203,349,571,890,898)(207,356,582,287,474)(209,360,589,410,385)(211,364,597,932,835)(213,368,604,225,386)(215,372,612,958,747)(217,376,619,970,628)(219,379,625,982,1466)(230,392,276,456,579)(234,397,318,519,538)(237,401,344,561,476)(239,404,359,587,388)(241,408,661,616,826)(246,414,403,355,580)(249,417,422,445,539)(252,420,437,508,477)(254,423,448,550,389)(256,427,684,733,941)(260,432,485,331,540)(263,435,500,430,478)(265,438,511,493,390)(267,442,703,820,899)(270,446,542,291,479)(272,449,553,411,391)(274,453,718,883,836)(278,460,729,925,748)(280,464,736,951,629)(282,468,744,1132,1459)(294,483,312,509,481)(296,486,338,551,394)(298,490,772,603,942)(302,495,399,333,543)(305,498,419,431,482)(307,501,434,494,395)(309,505,791,722,900)(314,512,497,412,396)(316,516,806,810,837)(320,523,817,872,749)(322,527,823,914,630)(324,531,831,1243,1226)(335,547,853,578,901)(340,554,416,413,400)(342,558,868,707,838)(346,565,879,795,750)(348,569,886,857,631)(350,573,894,1328,1398)(357,584,910,537,839)(361,591,921,688,751)(363,595,928,776,632)(365,599,936,1390,1336)(369,606,947,475,752)(371,610,954,665,633)(373,614,962,1431,1250)(375,617,889,387,634)(377,621,974,1451,1136)(380,626,984,975,1453)(393,641,463,734,944)(398,647,526,821,903)(402,652,568,884,841)(405,656,594,926,754)(407,659,609,952,636)(409,663,1028,979,1460)(415,670,658,605,945)(418,673,681,723,904)(421,676,700,811,842)(424,679,715,873,755)(426,682,726,915,637)(428,686,1054,1125,1235)(433,692,769,581,905)(436,695,788,708,843)(439,698,803,796,756)(441,701,814,858,638)(443,705,1076,1236,1399)(447,710,850,541,844)(450,713,865,689,757)(452,716,876,777,639)(454,720,1095,1321,1337)(457,724,907,480,758)(459,727,918,666,640)(461,731,1110,1383,1251)(465,738,1121,1424,1137)(467,742,1128,1444,987)(469,745,1134,1644,2068)(484,763,520,812,846)(487,767,562,874,760)(489,770,588,916,643)(491,774,1169,966,1242)(496,780,654,583,908)(499,783,678,709,847)(502,786,697,797,761)(504,789,712,859,644)(506,793,1192,1114,1400)(510,798,765,544,848)(513,801,785,690,762)(515,804,800,778,645)(517,808,1211,1227,1338)(522,815,862,667,646)(528,825,1233,1372,1138)(530,829,1239,1413,988)(532,833,1247,1777,2057)(546,851,552,860,649)(548,855,1274,943,1401)(555,863,675,691,766)(557,866,694,779,650)(559,870,1293,1099,1339)(564,877,782,668,651)(566,881,1308,1215,1252)(570,888,1318,1297,1139)(572,892,1324,1357,989)(574,896,1332,1881,1751)(585,912,1354,902,1340)(590,919,672,669,655)(592,923,1368,1080,1253)(596,930,1379,1196,1140)(598,934,1386,1278,990)(600,938,1394,1962,1970)(607,949,1409,840,1254)(611,956,1420,1058,1141)(613,960,1427,1173,991)(615,964,1435,2019,1888)(618,968,1389,753,1142)(620,972,1447,1031,992)(622,976,1455,2049,1781)(624,980,1462,635,993)(627,985,1471,1638,2247)(642,1001,741,1126,1320)(648,1008,828,1237,1402)(653,1014,891,1322,1342)(657,1019,933,1384,1256)(660,1023,959,1425,1144)(662,1026,971,1445,995)(664,1029,1518,1467,2069)(671,1037,1025,967,1327)(674,1040,1051,1115,1403)(677,1043,1073,1228,1343)(680,1046,1092,1312,1257)(683,1049,1107,1373,1145)(685,1052,1118,1414,996)(687,1056,1547,1641,2058)(693,1063,1166,946,1404)(696,1066,1189,1100,1344)(699,1069,1208,1216,1258)(702,1072,1223,1298,1146)(704,1074,1230,1358,997)(706,1078,1570,1770,1760)(711,1084,1271,906,1345)(714,1087,1290,1081,1259)(717,1090,1305,1197,1147)(719,1093,1315,1279,998)(721,1097,1592,1874,1971)(725,1102,1351,845,1260)(728,1105,1365,1059,1148)(730,1108,1376,1174,999)(732,1112,1611,1955,1889)(735,1116,1406,759,1149)(737,1119,1417,1032,1000)(739,1123,1626,2012,1782)(743,1130,1637,2043,1647)(746,1135,1646,2118,2046)(764,1155,822,913,1347)(768,1160,885,1313,1262)(771,11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705,233,396,645,1005,1490,2092,2762,1112,1612,2216,2698,604,944,1402,1974,2620,3269,3826,2563,543,847,1263,1794,2416,3066,3656,2324,401,651,1012,1497,2099,2766,3419,1628,2235,2889,3350,947,1406,1978,2624,3271,3827,4218,3215,851,1268,1800,2422,3072,3662,4106,2983,656,1018,1504,2105,2771,3146,3727,2251,2908,3525,3893,1409,1982,2628,547,854,1273,1807,2430,1272,1805,2427,3076,3664,3857,4230,3592,1023,1510,2112,2777,3425,3728,1218,1742,2361,3015,3614,1985,2632,3278,870,1294,1832,2457,3106,1809,2432,3081,3668,4108,4231,1947,2587,1515,2118,2784,3431,3959,4155,1746,259,431,690,1060,1552,2153,2817,1310,1853,2481,2713,3363,2436,3086,3673,4112,4377,4439,2590,561,873,1298,1838,2466,3116,3445,2369,435,19,37,70,129,230,393,642,1002,1487,2090,958,1424,2001,2649,3294,3841,4120,3241,877,97,174,299,492,776,1173,626,698,42,79,143,249,418,674,1041,1531,2133,2800,1427,2005,2653,3298,3844,4225,4384,2265,1306,177,303,497,782,1180,1698,1135,1072,84,152,263,436,696,1067,1560,2162,2823,3175,2008,2656,3300,3845,4226,4438,4504,2930,1850,306,500,785,1183,1701,2317,1780,1566,157,272,450,714,1088,1583,2186,2845,3478,3750,2659,3303,3847,1411,1986,2634,3281,3548,2480,503,788,1186,1704,2320,2976,2521,2169,277,459,728,1106,1604,2208,2866,3495,3999,4169,3306,3849,4227,1999,2648,3293,3840,4034,3131,791,1189,1707,2323,2979,3253,3267,2831,464,737,1120,1622,2227,2881,724,1101,1598,2202,2861,3490,3997,2661,3307,3851,3912,4268,3714,1192,1710,2326,2982,3588,3813,1322,1865,742,1129,1636,2245,2900,3517,1105,107,193,331,541,845,1261,1792,2414,3064,3655,2019,2672,1713,2329,2985,3591,4061,4211,1869,317,518,810,1215,1739,2358,2306,1606,198,340,555,864,1286,1823,2448,3098,3685,4123,2675,3319,2332,2988,3594,4064,4353,3600,2500,521,814,1220,1745,2365,3020,2990,2213,345,564,878,1304,1845,2473,3123,3705,4136,4188,3322,3860,2989,3595,4065,4354,4492,4068,1527,817,1223,1748,2368,3023,3622,3612,2872,569,887,1317,1860,2490,3141,3723,4151,4401,4419,2444,3094,3596,2738,3395,3934,4287,4356,2132,119,217,377,622,977,1458,2056,2724,892,1325,1870,2498,3148,3645,4094,4368,4497,4519,3097,752,1141,1651,2264,2923,3541,4029,2801,281,467,743,1131,1640,2252,2910,3529,1330,1878,2507,1340,1891,2524,1632,2240,2894,3513,3686,1149,1660,2271,2929,3545,4031,4338,3449,531,832,1246,1776,2397,3049,2512,3158,1883,2515,3162,1898,2531,3179,2244,368,605,946,1405,1668,2279,2934,2741,3398,3936,3332,3872,896,1333,1884,2517,3165,3738,3161,827,1236,1761,2380,2538,3186,3754,2902,610,955,1419,1993,2287,2942,3552,3402,3940,4290,3875,4242,1396,1967,2614,3264,3822,4215,3736,1239,1764,2383,3037,3193,3761,4173,3522,960,1428,2007,2655,2949,3558,4036,3944,4294,4468,4245,4443,2025,2684,3335,3878,4247,4446,2841,1767,2385,3039,3634,3766,4176,4411,4014,1433,2016,2668,3311,3564,4040,4340,4298,4471,4535,3720,4147,2723,3378,3919,4250,4448,4528,3477,774,1170,1687,2303,2961,3571,4047,4329,2021,2676,3321,3859,4046,4344,4488,4475,4538,4556,4150,1722,2339,2994,3599,4067,4355,4309,3991,1194,1714,2331,2987,3593,4063,4352,4484,2681,3329,3868,2573,3221,3784,2918,3536,4025,4335,4402,2343,414,669,1035,1524,2128,2795,3443,1737,2355,3009,3357,3898,4260,3742,4159,3334,3876,4244,3225,860,1281,1817,980,1463,2063,2732,3001,673,15,30,59,112,203,350,574,897,1335,1887,1415,1989,2637,1777,2398,3050,3642,3792,1285,165,285,472,749,1138,1648,2261,1043,46,87,158,274,454,721,1098,1595,2199,2858,1992,352,576,899,1337,1889,2523,3172,1825,290,478,756,1146,1657,2268,2926,1536,103,186,320,524,819,1226,1751,2371,3026,3625,2642,580,904,1343,1895,2529,3178,3747,2453,483,762,1153,1665,2277,2933,3547,2141,202,348,570,889,1320,1864,2493,3144,3726,4154,3289,908,1348,1901,2535,3184,3753,4166,3105,767,1159,1672,2284,2940,3551,4033,2809,364,598,935,1389,1954,2595,3243,3805,4205,4432,3839,1352,1906,2541,3190,3759,4172,4410,3692,1164,1678,2291,2946,3556,3807,4206,3456,614,963,1434,2018,2671,3315,1084,1578,2180,2838,3474,1910,2546,3195,3763,4174,4265,4452,4129,1683,2297,2953,3561,4038,4207,1857,2486,976,1456,2052,2688,3337,3879,1582,172,296,487,768,1161,1675,2288,2944,3555,4035,2664,3308,2302,2959,3568,4043,4342,4433,2489,509,797,1198,1719,2336,2991,2792,2188,308,504,790,1188,1706,2322,2978,3585,4057,4349,3310,926,1373,1932,2570,3219,3783,4052,3142,801,76,138,243,411,666,1032,1521,516,807,1210,1451,2043,2704,3354,3895,4258,4412,3854,1377,196,336,549,857,1278,1815,2083,1206,142)

Polycyclic presentation:

5,4561|b^a=b^1439

Conjugacy classes & character table

917 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22805, 1);