Finite Group DatabasebetaFinite Group Databasebeta| order | 20555 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C4111⋊C5 |
| Fitting, socle, derived, Sylow | C4111 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C4111 | C5 |
| LCS | G ▷ C4111 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 4111 |
| transitive degree | μt(G) = 4111 |
A minimal faithful permutation representation — 2 generators on 4111 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)(32,62,117,211,360)(37,69,41,76,68)(39,73,133,106,190)(43,80,144,148,164)(45,84,151,174,122)(47,88,159,273,349)(51,95,170,92,165)(53,99,177,137,123)(55,103,185,315,323)(57,107,192,67,124)(59,111,200,341,281)(61,115,207,353,219)(63,119,215,74,135)(70,128,83,149,167)(72,131,98,175,126)(77,139,130,94,168)(79,142,141,138,127)(81,146,251,266,324)(85,153,262,308,282)(87,157,269,334,220)(89,161,277,451,580)(96,172,293,191,325)(100,179,304,255,283)(102,183,311,297,221)(104,187,319,514,564)(108,194,330,166,284)(110,198,337,240,222)(112,202,345,556,522)(114,205,350,125,223)(116,209,356,573,459)(118,213,364,586,375)(120,217,371,244,404)(129,228,156,267,327)(132,232,182,309,286)(134,235,197,335,225)(136,238,239,361,480)(140,243,234,193,328)(143,246,248,256,287)(145,249,259,298,226)(147,253,381,444,368)(150,257,290,169,288)(152,260,301,241,227)(154,264,429,507,523)(158,271,440,549,460)(160,275,447,569,376)(162,279,455,382,237)(171,291,176,299,230)(173,295,477,216,369)(178,302,245,242,231)(180,306,492,433,524)(184,313,503,496,461)(186,317,510,538,377)(188,321,518,789,882)(195,332,534,326,525)(199,339,545,418,462)(201,343,552,481,378)(203,347,560,852,860)(206,351,566,285,463)(208,354,570,400,379)(210,358,576,875,797)(212,362,582,224,380)(214,366,590,897,710)(218,373,601,485,741)(229,386,274,445,565)(233,391,316,508,527)(236,395,342,550,465)(247,407,412,434,528)(250,410,426,497,466)(252,413,437,539,383)(254,416,638,696,736)(258,421,474,329,529)(261,424,489,419,467)(263,427,500,482,384)(265,431,666,782,594)(268,435,531,289,468)(270,438,542,401,385)(272,442,681,845,798)(276,449,692,871,711)(278,453,699,890,605)(280,457,706,639,636)(292,472,310,498,470)(294,475,336,540,388)(296,479,734,581,755)(300,484,393,331,532)(303,487,409,420,471)(305,490,423,483,389)(307,494,752,685,595)(312,501,486,402,390)(314,505,767,771,799)(318,512,778,834,712)(320,516,785,866,606)(322,520,793,716,397)(333,536,815,370,596)(338,543,406,403,394)(340,547,830,670,800)(344,554,841,756,713)(346,558,848,819,607)(348,562,856,1229,1266)(352,567,863,526,801)(355,571,868,651,714)(357,574,872,737,608)(359,578,878,1259,1237)(363,584,886,464,715)(365,588,893,632,609)(367,592,901,1294,1150)(372,599,422,655,731)(374,603,917,823,1181)(387,616,452,697,884)(392,622,515,783,861)(396,627,557,846,803)(398,417,587,891,611)(399,583,818,405,637)(408,640,610,686,598)(411,643,663,772,804)(414,646,678,835,717)(415,617,689,703,612)(425,658,749,671,805)(428,661,764,757,718)(430,664,775,820,613)(432,668,983,1136,1070)(436,673,812,530,806)(439,676,827,652,719)(441,679,838,738,614)(443,683,1001,1222,905)(446,687,862,469,720)(448,690,867,633,615)(450,694,1016,1255,1151)(454,701,1027,1287,1041)(456,704,618,478,732)(458,708,1037,971,1075)(473,725,509,773,808)(476,729,551,836,722)(488,744,645,672,809)(491,747,660,758,723)(493,750,675,821,619)(495,754,1089,1020,1090)(499,759,727,533,810)(502,762,746,653,724)(504,765,761,739,620)(506,769,1106,1125,906)(511,776,824,634,621)(513,780,1121,1211,1152)(517,787,1132,1250,1042)(519,791,1139,1276,921)(521,795,1146,1077,950)(535,813,541,822,624)(537,817,1174,883,1109)(544,825,642,654,728)(546,828,657,740,625)(548,832,1192,1005,907)(553,839,743,635,626)(555,843,1207,1110,1153)(559,850,1218,1196,1043)(561,854,1225,1244,922)(563,858,1233,1156,629)(568,864,1241,597,908)(572,869,1247,986,1154)(575,873,1252,1091,1044)(577,876,1256,1177,923)(579,880,1263,1709,1714)(585,888,1272,802,1155)(589,895,1283,965,1045)(591,899,1290,1071,924)(593,903,1298,1758,1677)(600,912,674,990,1171)(602,915,760,1095,1063)(604,919,1320,1034,1435)(623,936,790,1137,1268)(628,942,853,1223,1238)(630,669,898,1288,1046)(631,887,1243,742,1076)(641,954,952,1021,1176)(644,957,980,1126,910)(647,960,998,1212,1157)(648,937,1013,1143,1047)(649,955,1024,1277,926)(650,894,1281,953,951)(656,970,1067,885,1195)(659,972,1086,1006,911)(662,975,1103,1111,1158)(665,978,1118,1197,1048)(667,981,1129,1031,927)(677,993,1189,987,1159)(680,996,1204,1092,1049)(682,999,1215,1178,928)(684,1003,1397,1663,1474)(688,1008,1240,807,1160)(691,1011,1246,966,1050)(693,1014,1251,1072,929)(695,1018,1307,1702,1302)(698,1022,1270,721,1051)(700,1025,1280,947,930)(702,1029,1428,1751,1568)(705,1033,938,816,1172)(707,1035,956,969,1064)(709,1039,1441,1385,1602)(726,1056,784,1127,1239)(730,1061,847,1213,1162)(733,1065,794,1144,1052)(735,1068,892,1278,931)(745,1078,925,1007,914)(748,1081,977,1112,1163)(751,1084,995,1198,1053)(753,1087,1010,1032,932)(763,1098,1083,988,1164)(766,1101,1100,1093,1054)(768,1104,1115,1179,933)(770,1108,1511,1547,1492)(774,1113,1167,811,1165)(777,1116,1186,967,1055)(779,1119,1201,1073,934)(781,1123,1528,1652,1303)(786,1130,1245,948,935)(788,1134,1543,1697,1569)(792,1141,1554,1740,1444)(796,1148,1564,1499,1478)(814,1169,837,1199,1058)(826,1184,959,989,1168)(829,1187,974,1094,1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02,2859,3260,1883,356,520,794,1145,1561,2019,2252,2736,3168,3516,3769,3934,3530,2731,3161,3508,3761,3927,4025,3697,2100,2562,2989,3351,3635,3840,3868,2551,747,69,127,226,383,612,926,1068,1471,1926,2401,2858,1428,1882,321,519,792,1142,1557,1758,2251,2735,3167,3515,3768,3179,3529,3782,3945,4038,4084,4086,3432,1640,2099,2561,2988,3350,3634,3677,2089,490,36,68,126,225,382,611,734,1067,1470,1925,2400,2857,1812,187,320,517,788,1135,1546,2001,2468,2909,3294,3600,2744,3178,3527,3779,3942,4035,4041,3081,1213,1639,2098,2560,2987,3349,3406,1631,304,489,746,1080,1482,1935,970,1363,1820,2308,2789,3217,3269,1354,103,186,318,513,781,1124,1531,1982,2448,2891,3280,2256,2743,3176,3524,3776,3939,3950,2653,846,1212,1638,2097,2559,2986,3050,1207,1630,2088,2550,2978,3342,2437,655,969,1362,1819,2307,2788,2874,960,54,102,184,314,506,770,1109,1512,1962,2430,2877,1760,2255,2741,3173,3521,3773,3788,2182,556,845,1211,1637,2096,2558,2644,3071,3422,3688,3874,3991,3649,1970,421,654,968,1361,1818,2306,2427,646,27,53,100,180,307,495,755,1090,1492,1944,2416,1298,1759,2253,2738,3170,3518,3536,1705,2181,2652,3080,3431,3696,3783,3946,4039,4085,4104,4110,4067,3371,1520,257,420,653,967,1360,1817,1959,413,13,26,51,96,173,296,480,736,1070,1474,1930,2406,1416,1870,2350,2818,3233,3187,3535,1775,2272,2761,3197,3544,3795,3915,4020,4075,4100,4109,4008,3014,1116,149,256,419,652,966,1359,1509,251,412,645,959,1353,1811,754,360,580,882,1266,1714,2196,1018,1415,1868,2347,2815,3230,3478,1311,1774,2271,2760,3196,3543,3738,3914,4019,4074,4099,3899,2588,776,82,148,255,418,651,965,1106,1508,1958,2426,2873,3268,2323,494,753,1088,1490,1942,2414,2793,694,1017,1414,1867,2346,2814,3126,912,1310,1773,2270,2759,3195,3475,3737,3913,4018,4073,3719,2124,510,775,1115,1519,1969,2436,2576,3002,3361,3642,3844,3975,3595,1841,306,493,751,1085,1487,1939,2315,449,693,1015,1411,1864,2343,2696,599,911,1309,1772,2269,2758,3120,3474,3736,3912,4017,3456,1659,2123,2587,3013,3370,3648,2915,3299,3604,3825,3968,4050,4056,3288,1389,179,305,491,748,1082,1484,1830,275,448,691,1012,1408,1861,2218,371,598,910,1308,1771,2268,2687,3119,3473,3735,3911,3103,3455,2041,2507,2944,3320,3617,2473,2914,3298,3603,3824,3967,3985,2903,996,99,178,303,488,745,1079,1376,159,274,446,688,1009,1405,1734,350,565,861,1238,1678,2151,2208,2686,3118,3472,3734,3910,3250,1583,2040,2506,2943,3319,2005,2472,2913,3297,3602,3823,3862,2462,679,52,98,176,300,485,742,983,1375,1829,2314,2792,3218,2113,1283,1733,2217,2695,3125,3477,2488,2929,3310,3612,3830,3970,4023,2841,1165,1582,2039,2505,2942,1549,2004,2471,2912,3296,3601,3668,1996,440,678,995,1388,1840,2322,1242,1322,1790,2290,2780,3215,3559,1650,2112,2575,3001,3360,3641,3757,2018,2487,2928,3309,3611,3829,3922,2378,810,1164,1581,2038,2504,1137,1548,2003,2470,2911,3295,3394,1543,1995,2461,2902,3287,3594,2825,864,919,1321,1788,2287,2776,3210,1220,1649,2110,2572,2998,3357,3503,1560,2017,2486,2927,3308,3610,3752,1898,532,809,1163,1580,2037,789,1136,1547,2002,2469,2910,3037,3393,1918,2393,2852,3256,3578,2358,567,603,918,1319,1785,2283,2771,850,1219,1647,2107,2569,2995,3156,1144,1559,2016,2485,2926,3307,3496,1441,330,531,808,1162,1579,2036,1076,1479,1934,2409,2864,3264,3337,1463,1917,2392,2851,3255,1878,351,373,602,916,1316,1781,2278,558,849,1217,1644,2104,2566,2726,793,1143,1558,2015,2484,2925,3148,1247,1687,2160,2629,3057,3410,2546,741,1075,1478,1933,2408,2863,2968,1061,1462,1916,2391,2850,1423,205,217,372,600,913,1312,1776,345,557,847,1214,1641,2101,2245,573,871,1250,1691,2164,2633,2718,3147,3495,3751,3921,4022,3674,2083,484,740,1074,1477,1932,2407,2536,729,1060,1461,1915,2390,1025,113,119,216,370,597,909,1307,1770,2267,2757,3194,3542,2720,1754,2244,2725,3155,3502,3756,3105,3459,3723,3902,4011,4069,4081,3402,1625,299,483,739,1073,1476,1931,2072,475,728,1059,1460,1914,699,1024,62,118,214,367,593,904,1301,1764,2262,2753,3191,2238,2719,3149,3497,3753,3923,3947,2671,3104,3457,3720,3900,4009,4034,3046,1202,175,298,482,738,1072,1475,1614,293,474,727,1058,1459,1913,954,31,61,116,210,359,579,881,1265,1713,2195,2673,1749,2237,2717,3146,3494,3750,3784,2193,2670,3102,3454,3718,3898,3941,2617,839,97,174,297,481,737,1071,1192,1613,2071,2535,2967,3336,2422,640,15,30,59,112,203,348,563,859,1236,1676,2150,1292,1748,2235,2714,3143,3491,3531,1711,2192,2668,3099,3452,3716,3778,2147,552,838,1201,1624,2082,2545,2638,3065,3417,3684,3872,3990,3638,1953,407,7,14,28,55,104,188,322,521,796,1149,1567,899,1291,1746,2232,2711,3140,3182,1263,1710,2190,2665,3097,3450,3526,1699,2173,2642,3069,3420,3686,3009,3367,3647,3847,3977,4052,4062,3356,1503,246,3,6,12,24,47,89,162,280,458,709,1040,590,898,1289,1743,2229,2708,2747,3181,1300,1763,2261,2752,3190,3177,3525,3777,3940,4033,4080,3896,2582,3008,3366,3646,3846,3976,3999,2997,1101,142)Polycyclic presentation:
5,4111|b^a=b^4070827 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20555, 1);