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Aliases
Splittings
Subgroups
4 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 6037 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,477)(382,430,474)(385,433,471)(388,436,468)(391,439,465)(394,442,462)(397,445,459)(400,448,456)(403,451,453)(406,454,450)(409,457,447)(412,460,444)(415,463,441)(418,466,438)(421,469,435)(424,472,432)(426,475,429)(428,479,532)(431,482,529)(434,485,526)(437,488,523)(440,491,520)(443,494,517)(446,497,514)(449,500,511)(452,503,508)(455,506,505)(458,509,502)(461,512,499)(464,515,496)(467,518,493)(470,521,490)(473,524,487)(476,527,484)(478,530,481)(480,534,590)(483,537,587)(486,540,584)(489,543,581)(492,546,578)(495,549,575)(498,552,572)(501,555,569)(504,558,566)(507,561,563)(510,564,560)(513,567,557)(516,570,554)(519,573,551)(522,576,548)(525,579,545)(528,582,542)(531,585,539)(533,588,536)(535,592,651)(538,595,648)(541,598,645)(544,601,642)(547,604,639)(550,607,636)(553,610,633)(556,613,630)(559,616,627)(562,619,624)(565,622,621)(568,625,618)(571,628,615)(574,631,612)(577,634,609)(580,637,606)(583,640,603)(586,643,600)(589,646,597)(591,649,594)(593,653,715)(596,656,712)(599,659,709)(602,662,706)(605,665,703)(608,668,700)(611,671,697)(614,674,694)(617,677,691)(620,680,688)(623,683,685)(626,686,682)(629,689,679)(632,692,676)(635,695,673)(638,698,670)(641,701,667)(644,704,664)(647,707,661)(650,710,658)(652,713,655)(654,717,782)(657,720,779)(660,723,776)(663,726,773)(666,729,770)(669,732,767)(672,735,764)(675,738,761)(678,741,758)(681,744,755)(684,747,752)(687,750,749)(690,753,746)(693,756,743)(696,759,740)(699,762,737)(702,765,734)(705,768,731)(708,771,728)(711,774,725)(714,777,722)(716,780,719)(718,784,852)(721,787,849)(724,790,846)(727,793,843)(730,796,840)(733,799,837)(736,802,834)(739,805,831)(742,808,828)(745,811,825)(748,814,822)(751,817,819)(754,820,816)(757,823,813)(760,826,810)(763,829,807)(766,832,804)(769,835,801)(772,838,798)(775,841,795)(778,844,792)(781,847,789)(783,850,786)(785,854,925)(788,857,922)(791,860,919)(794,863,916)(797,866,913)(800,869,910)(803,872,907)(806,875,904)(809,878,901)(812,881,898)(815,884,895)(818,887,892)(821,890,889)(824,893,886)(827,896,883)(830,899,880)(833,902,877)(836,905,874)(839,908,871)(842,911,868)(845,914,865)(848,917,862)(851,920,859)(853,923,856)(855,927,1001)(858,930,998)(861,933,995)(864,936,992)(867,939,989)(870,942,986)(873,945,983)(876,948,980)(879,951,977)(882,954,974)(885,957,971)(888,960,968)(891,963,965)(894,966,962)(897,969,959)(900,972,956)(903,975,953)(906,978,950)(909,981,947)(912,984,944)(915,987,941)(918,990,938)(921,993,935)(924,996,932)(926,999,929)(928,1003,1080)(931,1006,1077)(934,1009,1074)(937,1012,1071)(940,1015,1068)(943,1018,1065)(946,1021,1062)(949,1024,1059)(952,1027,1056)(955,1030,1053)(958,1033,1050)(961,1036,1047)(964,1039,1044)(967,1042,1041)(970,1045,1038)(973,1048,1035)(976,1051,1032)(979,1054,1029)(982,1057,1026)(985,1060,1023)(988,1063,1020)(991,1066,1017)(994,1069,1014)(997,1072,1011)(1000,1075,1008)(1002,1078,1005)(1004,1082,1162)(1007,1085,1159)(1010,1088,1156)(1013,1091,1153)(1016,1094,1150)(1019,1097,1147)(1022,1100,1144)(1025,1103,1141)(1028,1106,1138)(1031,1109,1135)(1034,1112,1132)(1037,1115,1129)(1040,1118,1126)(1043,1121,1123)(1046,1124,1120)(1049,1127,1117)(1052,1130,1114)(1055,1133,1111)(1058,1136,1108)(1061,1139,1105)(1064,1142,1102)(1067,1145,1099)(1070,1148,1096)(1073,1151,1093)(1076,1154,1090)(1079,1157,1087)(1081,1160,1084)(1083,1164,1247)(1086,1167,1244)(1089,1170,1241)(1092,11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425,2546,2670,2797,2927,3060,3196,3335,3477,3622,3770,3920,4070,4215,4354,4487,4614,4735,4850,4959,5062,5159,5250,5335,5414,5452,5286,5097,4884,4647,4386,4101,3799,3503,3219,2947,2687,2439,2203,1979,1767,1567,1379,1203,1039,887,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,934,1010,1089,1171,1256,1344,1435,1529,1626,1726,1829,1935,2044,2156,2271,2389,2510,2634,2761,2891,3024,3160,3299,3441,3586,3734,3884,4034,4119,3817,3521,3237,2965,2705,2457,2221,1997,1785,1585,1397,1221,1057,905,765,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,746,813,883,956,1032,1111,1193,1278,1366,1457,1551,1648,1748,1851,1957,2066,2178,2293,2411,2532,2656,2783,2913,3046,3182,3321,3463,3608,3756,3906,4056,4201,4340,4473,4600,4721,4836,4945,5048,5146,5116,4907,4674,4417,4136,3835,3539,3255,2983,2723,2475,2239,2015,1803,1603,1415,1239,1075,923,782,849,919,992,1068,1147,1229,1314,1402,1493,1587,1684,1784,1887,1993,2102,2214,2329,2447,2568,2692,2819,2949,3082,3218,3357,3499,3644,3792,3942,4091,4235,4179,3880,3579,3726,3875,4025,4173,4315,4451,4581,4705,4823,4935,5041,5141,5235,5323,5405,5481,5551,5615,5673,5725,5617,5480,5319,5134,4925,4692,4435,4154,3853,3557,3273,3001,2741,2493,2257,2033,1820,1923,2029,2138,2250,2365,2483,2604,2728,2855,2985,3118,3254,3393,3535,3680,3828,3978,4126,4268,4406,4538,4664,4784,4898,5006,5108,5204,5294,5378,5456,5528,5487,5331,5152,4949,4722,4471,4196,3898,3597,3307,3029,2763,2509,2267,2037,2148,2261,2376,2494,2615,2739,2866,2996,3129,3265,3404,3546,3691,3839,3989,4137,4279,4415,4545,4669,4453,4172,3871,3575,3290,3429,3571,3716,3864,4014,4162,4304,4440,4570,4694,4812,4924,5030,5130,5224,5312,5394,5470,5540,5606,5666,5720,5768,5810,5846,5878,5907,5933,5956,5976,5993,5981,5939,5886,5821,5739,5633,5503,5349,5170,4967,4740,4489,4214,3916,3615,3325,3047,2781,2527,2285,2055,1837,1631,1437,1255,1085,927,1002,1079,1158,1240,1325,1413,1504,1598,1695,1795,1898,2004,2113,2225,2340,2458,2579,2703,2830,2960,3093,3229,3368,3510,3655,3803,3953,4102,4246,4384,4516,4642,4762,4877,4986,5089,5186,5277,5362,5441,5514,5581,5642,5697,5746,5789,5826,5859,5889,5916,5940,5951,5898,5833,5751,5645,5515,5361,5183,4981,4755,4505,4231,3934,3633,3343,3065,2799,2545,2303,2073,1855,1649,1455,1273,1103,945,799,665,543,433,335,249,288,329,372,418,467,519,574,632,693,757,824,894,967,1043,1122,1204,1289,1377,1468,1562,1659,1759,1862,1968,2077,2189,2304,2422,2543,2667,2794,2924,3057,3193,3332,3474,3619,3767,3917,4067,4212,4351,4484,4611,4732,4847,4956,5059,5156,5247,5332,5373,5196,4995,4770,4521,4248,3952,3651,3361,3083,2817,2563,2321,2091,1873,1667,1473,1291,1121,963,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,858,931,1007,1086,1168,1253,1341,1432,1526,1623,1723,1826,1932,2041,2153,2268,2386,2507,2631,2758,2888,3021,3157,3296,3438,3583,3731,3881,3970,3669,3379,3101,2835,2581,2339,2109,1891,1685,1491,1309,1139,981,835,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,749,816,886,959,1035,1114,1196,1281,1369,1460,1554,1651,1751,1854,1960,2069,2181,2296,2414,2535,2659,2786,2916,3049,3185,3324,3466,3611,3759,3909,4059,4204,4343,4476,4603,4724,4839,4948,5051,5148,5240,5213,5016,4795,4550,4281,3988,3687,3397,3119,2853,2599,2357,2127,1909,1703,1509,1327,1157,999,852,922,995,1071,1150,1232,1317,1405,1496,1590,1687,1787,1890,1996,2105,2217,2332,2450,2571,2695,2822,2952,3085,3221,3360,3502,3647,3795,3945,4094,4238,4376,4321,4033,3730,3434,3578,3724,3872,4022,4170,4312,4448,4578,4702,4820,4932,5038,5138,5232,5320,5402,5478,5548,5612,5670,5553,5404,5231,5034,4813,4568,4299,4006,3705,3415,3137,2871,2617,2375,2145,1926,2032,2141,2253,2368,2486,2607,2731,2858,2988,3121,3257,3396,3538,3683,3831,3981,4129,4271,4407,4539,4665,4785,4899,5007,5109,5205,5295,5379,5457,5529,5595,5557,5413,5246,5055,4840,4601,4338,4051,3748,3452,3168,2896,2636,2388,2152,1928,2036,2146,2258,2373,2491,2612,2736,2863,2993,3126,3262,3401,3543,3688,3836,3986,4134,4276,4412,4542,4317,4024,3723,3432,3574,3719,3867,4017,4165,4307,4443,4573,4697,4815,4927,5033,5133,5227,5315,5397,5473,5543,5607,5667,5721,5769,5811,5847,5879,5908,5934,5957,5977,5994,6008,5998,5962,5915,5856,5784,5690,5572,5430,5264,5073,4858,4619,4356,4069,3766,3470,3186,2914,2654,2406,2170,1946,1734,1534,1346,1170,1006,854,926,1000,1076,1155,1237,1322,1410,1501,1595,1692,1792,1895,2001,2110,2222,2337,2455,2576,2700,2827,2957,3090,3226,3365,3507,3652,3800,3950,4099,4243,4381,4513,4639,4760,4875,4984,5087,5184,5275,5360,5439,5512,5579,5640,5695,5744,5787,5824,5857,5887,5914,5927,5868,5796,5702,5584,5442,5276,5086,4872,4634,4372,4086,3784,3488,3204,2932,2672,2424,2188,1964,1752,1552,1364,1188,1024,872,732,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,754,821,891,964,1040,1119,1201,1286,1374,1465,1559,1656,1756,1859,1965,2074,2186,2301,2419,2540,2664,2791,2921,3054,3190,3329,3471,3616,3764,3914,4064,4209,4348,4481,4608,4729,4844,4953,5056,5153,5244,5289,5100,4887,4650,4389,4104,3802,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)
Polycyclic presentation:
3,6037|b^a=b^5527
Conjugacy classes & character table
2015 conjugacy classes. The character table is too large to display.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18111, 1);
Finite Group Databasebeta