#34095280

order	20036
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₅₀₀₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅₀₀₉	C₄
prime radical, cosocle	D₅₀₀₉	C₂
LCS	G ▷ C₅₀₀₉
UCS	C₁

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 5009
transitive degree	μ_t(G) = 5009

A minimal faithful permutation representation — 2 generators on 5009 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,190)(128,159,160,186)(132,163,164,182)(136,167,168,178)(140,171,172,174)(144,175,176,170)(148,179,180,166)(152,183,184,162)(154,187,188,158)(157,192,193,231)(161,196,197,227)(165,200,201,223)(169,204,205,219)(173,208,209,215)(177,212,213,211)(181,216,217,207)(185,220,221,203)(189,224,225,199)(191,228,229,195)(194,233,234,276)(198,237,238,272)(202,241,242,268)(206,245,246,264)(210,249,250,260)(214,253,254,256)(218,257,258,252)(222,261,262,248)(226,265,266,244)(230,269,270,240)(232,273,274,236)(235,278,279,325)(239,282,283,321)(243,286,287,317)(247,290,291,313)(251,294,295,309)(255,298,299,305)(259,302,303,301)(263,306,307,297)(267,310,311,293)(271,314,315,289)(275,318,319,285)(277,322,323,281)(280,327,328,378)(284,331,332,374)(288,335,336,370)(292,339,340,366)(296,343,344,362)(300,347,348,358)(304,351,352,354)(308,355,356,350)(312,359,360,346)(316,363,364,342)(320,367,368,338)(324,371,372,334)(326,375,376,330)(329,380,381,435)(333,384,385,431)(337,388,389,427)(341,392,393,423)(345,396,397,419)(349,400,401,415)(353,404,405,411)(357,408,409,407)(361,412,413,403)(365,416,417,399)(369,420,421,395)(373,424,425,391)(377,428,429,387)(379,432,433,383)(382,437,438,496)(386,441,442,492)(390,445,446,488)(394,449,450,484)(398,453,454,480)(402,457,458,476)(406,461,462,472)(410,465,466,468)(414,469,470,464)(418,473,474,460)(422,477,478,456)(426,481,482,452)(430,485,486,448)(434,489,490,444)(436,493,494,440)(439,498,499,561)(443,502,503,557)(447,506,507,553)(451,510,511,549)(455,514,515,545)(459,518,519,541)(463,522,523,537)(467,526,527,533)(471,530,531,529)(475,534,535,525)(479,538,539,521)(483,542,543,517)(487,546,547,513)(491,550,551,509)(495,554,555,505)(497,558,559,501)(500,563,564,630)(504,567,568,626)(508,571,572,622)(512,575,576,618)(516,579,580,614)(520,583,584,610)(524,587,588,606)(528,591,592,602)(532,595,596,598)(536,599,600,594)(540,603,604,590)(544,607,608,586)(548,611,612,582)(552,615,616,578)(556,619,620,574)(560,623,624,570)(562,627,628,566)(565,632,633,703)(569,636,637,699)(573,640,641,695)(577,644,645,691)(581,648,649,687)(585,652,653,683)(589,656,657,679)(593,660,661,675)(597,664,665,671)(601,668,669,667)(605,672,673,663)(609,676,677,659)(613,680,681,655)(617,684,685,651)(621,688,689,647)(625,692,693,643)(629,696,697,639)(631,700,701,635)(634,705,706,780)(638,709,710,776)(642,713,714,772)(646,717,718,768)(650,721,722,764)(654,725,726,760)(658,729,730,756)(662,733,734,752)(666,737,738,748)(670,741,742,744)(674,745,746,740)(678,749,750,736)(682,753,754,732)(686,757,758,728)(690,761,762,724)(694,765,766,720)(698,769,770,716)(702,773,774,712)(704,777,778,708)(707,782,783,861)(711,786,787,857)(715,790,791,853)(719,794,795,849)(723,798,799,845)(727,802,803,841)(731,806,807,837)(735,810,811,833)(739,814,815,829)(743,818,819,825)(747,822,823,821)(751,826,827,817)(755,830,831,813)(759,834,835,809)(763,838,839,805)(767,842,843,801)(771,846,847,797)(775,850,851,793)(779,854,855,789)(781,858,859,785)(784,863,864,946)(788,867,868,942)(792,871,872,938)(796,875,876,934)(800,879,880,930)(804,883,884,926)(808,887,888,922)(812,891,892,918)(816,895,896,914)(820,899,900,910)(824,903,904,906)(828,907,908,902)(832,911,912,898)(836,915,916,894)(840,919,920,890)(844,923,924,886)(848,927,928,882)(852,931,932,878)(856,935,936,874)(860,939,940,870)(862,943,944,866)(865,948,949,1035)(869,952,953,1031)(873,956,957,1027)(877,960,961,1023)(881,964,965,1019)(885,968,969,1015)(889,972,973,1011)(893,976,977,1007)(897,980,981,1003)(901,984,985,999)(905,988,989,995)(909,992,993,991)(913,996,997,987)(917,1000,1001,983)(921,1004,1005,979)(925,1008,1009,975)(929,1012,1013,971)(933,1016,1017,967)(937,1020,1021,963)(941,1024,1025,959)(945,1028,1029,955)(947,1032,1033,951)(950,1037,1038,1128)(954,1041,1042,1124)(958,1045,1046,1120)(962,1049,1050,1116)(966,1053,1054,1112)(970,1057,1058,1108)(974,1061,1062,1104)(978,1065,1066,1100)(982,1069,1070,1096)(986,1073,1074,1092)(990,1077,1078,1088)(994,1081,1082,1084)(998,1085,1086,1080)(1002,1089,1090,1076)(1006,1093,1094,1072)(1010,1097,1098,1068)(1014,1101,1102,1064)(1018,1105,1106,1060)(1022,1109,1110,1056)(1026,1113,1114,1052)(1030,1117,1118,1048)(1034,1121,1122,1044)(1036,1125,1126,1040)(1039,1130,1131,1225)(1043,1134,1135,1221)(1047,1138,1139,1217)(1051,1142,1143,1213)(1055,1146,1147,1209)(1059,1150,1151,1205)(1063,1154,1155,1201)(1067,1158,1159,1197)(1071,1162,1163,1193)(1075,1166,1167,1189)(1079,1170,1171,1185)(1083,1174,1175,1181)(1087,1178,1179,1177)(1091,1182,1183,1173)(1095,1186,118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1,1502,1397,1296,1199,1106,1017,932,851,774,701,630,699,772,849,930,1015,1104,1197,1294,1395,1500,1609,1722,1839,1960,2085,2214,2347,2484,2625,2770,2919,3072,3229,3390,3555,3724,3897,4074,4250,4360,4430,4496,4558,4616,4670,4720,4766,4808,4846,4880,4910,4936,4958,4976,4990,5000,5003,4996,4985,4970,4951,4928,4901,4870,4835,4796,4753,4706,4655,4600,4541,4478,4411,4340,4186,4338,4269,4107,3926,3749,3576,3407,3242,3081,2924,2771,2622,2477,2336,2199,2066,1937,1812,1691,1574,1461,1352,1247,1146,1049,956,867,782,862,945,1030,1119,1212,1309,1410,1515,1624,1737,1854,1975,2100,2229,2362,2499,2640,2785,2934,3087,3244,3405,3570,3739,3912,4089,4221,4040,3863,3690,3521,3356,3195,3038,2885,2736,2591,2450,2313,2180,2051,1926,1805,1688,1575,1466,1361,1260,1163,1070,981,896,815,738,665,596,531,470,413,360,311,266,225,188,153,186,223,264,309,358,411,468,529,594,663,736,813,894,979,1068,1161,1258,1359,1464,1573,1686,1803,1924,2049,2178,2311,2448,2589,2734,2883,3036,3193,3354,3519,3688,3861,4038,4219,4351,4421,4487,4549,4607,4661,4711,4757,4769,4726,4679,4628,4573,4514,4451,4384,4304,4152,3975,3802,3633,3468,3307,3150,2997,2848,2701,2846,2995,3148,3305,3466,3631,3800,3973,4150,4302,4251,4071,3890,3713,3540,3371,3206,3045,2888,2735,2586,2441,2300,2163,2030,1901,1776,1655,1771,1890,2011,2136,2265,2398,2535,2676,2821,2970,3123,3280,3441,3606,3775,3948,4125,4277,4364,4339,4183,4002,3825,3652,3483,3318,3157,3000,2847,2698,2553,2412,2275,2142,2013,1888,1767,1650,1537,1428,1323,1222,1125,1032,943,858,777,700,627,558,493,432,375,322,273,228,187,150,117,88,63,42,25,12,3,7,14,23,36,53,74,99,128,161,198,239,284,333,386,443,504,569,638,711,788,869,954,1043,1136,1233,1334,1439,1548,1661,1778,1899,2024,2153,2286,2423,2564,2709,2858,3011,3168,3329,3494,3663,3836,4013,4194,4116,3939,3766,3597,3432,3271,3114,2961,2812,2667,2526,2389,2256,2127,2002,1881,1764,1651,1540,1649,1762,1879,2000,2125,2254,2387,2524,2665,2810,2959,3112,3269,3430,3595,3764,3937,4114,4270,4220,4035,3854,3677,3504,3335,3170,3009,2852,3004,3159,3316,3477,3642,3811,3984,4161,4313,4391,4458,4521,4580,4635,4686,4733,4776,4815,4850,4834,4792,4746,4696,4642,4584,4522,4456,4386,4303,4147,3966,3789,3616,3447,3282,3121,2964,2811,2662,2517,2376,2239,2106,1977,1852,1731,1614,1501,1392,1287,1186,1089,996,907,822,741,664,591,522,457,396,339,286,237,192,232,275,320,369,422,479,540,605,674,747,824,905,990,1079,1172,1269,1370,1475,1584,1697,1814,1935,2060,2189,2322,2459,2600,2745,2894,3047,3204,3365,3530,3699,3872,4049,4230,4080,3903,3730,3561,3396,3235,3078,2925,2776,2631,2490,2353,2220,2091,1966,1845,1728,1615,1506,1401,1300,1203,1110,1021,936,855,778,703,776,853,934,1019,1108,1201,1298,1399,1504,1613,1726,1843,1964,2089,2218,2351,2488,2629,2774,2923,3076,3233,3394,3559,3728,3901,4078,4252,4361,4431,4497,4559,4617,4671,4721,4767,4809,4847,4881,4911,4937,4959,4977,4991,5001,5007,5009,5005,4998,4987,4972,4953,4930,4903,4872,4837,4798,4755,4708,4657,4602,4543,4480,4413,4342,4271,4111,3930,3753,3580,3411,3246,3085,2928,2775,2626,2481,2340,2203,2070,1941,1816,1695,1578,1465,1356,1251,1150,1053,960,871,786,705,781,860,941,1026,1115,1208,1305,1406,1511,1620,1733,1850,1971,2096,2225,2358,2495,2636,2781,2930,3083,3240,3401,3566,3735,3908,4085,4225,4044,3867,3694,3525,3360,3199,3042,2889,2740,2595,2454,2317,2184,2055,1930,1809,1692,1579,1470,1365,1264,1167,1074,985,900,819,742,669,600,535,474,417,364,315,270,229,190,227,268,313,362,415,472,533,598,667,740,817,898,983,1072,1165,1262,1363,1468,1577,1690,1807,1928,2053,2182,2315,2452,2593,2738,2887,3040,3197,3358,3523,3692,3865,4042,4223,4352,4422,4488,4550,4608,4662,4712,4758,4800,4811,4772,4729,4682,4631,4576,4517,4454,4387,4308,4156,3979,3806,3637,3472,3311,3154,3001,2850,2999,3152,3309,3470,3635,3804,3977,4154,4306,4253,4075,3894,3717,3544,3375,3210,3049,2892,2739,2590,2445,2304,2167,2034,1905,1780,1659,1542,1654,1769,1886,2007,2132,2261,2394,2531,2672,2817,2966,3119,3276,3437,3602,3771,3944,4121,4189,4008,3831,3658,3489,3324,3163,3006,2853,2704,2559,2418,2281,2148,2019,1894,1773,1656,1543,1434,1329,1228,1131,1038,949,864,783,706,633,564,499,438,381,328,279,234,193,156,123,94,69,48,31,18,9,4)

Polycyclic presentation:

2^2,5009|a^2=b,c^a=c^4470,c^b=c^5008

Conjugacy classes & character table

1256 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20036, 3);