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Aliases
Splittings
Subgroups
4 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 6343 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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69,5346,5420,5491,5559,5624,5686,5745,5801,5856,5908,5957,6003,6046,6086,6123,6157,6131,6055,5967,5867,5755,5631,5495,5347,5187,5015,4831,4635,4424,4189,3930,3647,3358,3080,2814,2560,2318,2088,1870,1664,1470,1288,1118,960,814,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,727,794,864,937,1013,1092,1174,1259,1347,1438,1532,1629,1729,1832,1938,2047,2159,2274,2392,2513,2637,2764,2894,3027,3163,3302,3444,3588,3732,3871,3882,3596,3307,3029,2763,2509,2267,2037,2148,2261,2376,2494,2615,2739,2866,2996,3129,3265,3404,3546,3690,3833,3970,4101,4226,4345,4458,4565,4666,4764,4859,4951,5040,5126,5209,5289,5366,5440,5511,5579,5644,5706,5765,5821,5874,5924,5971,6015,6056,6094,6040,5947,5842,5725,5596,5455,5302,5137,4959,4769,4567,4344,4097,3826,3536,3252,2980,2720,2472,2236,2012,1800,1600,1412,1236,1072,920,780,651,712,776,843,913,986,1062,1141,1223,1308,1396,1487,1581,1678,1778,1881,1987,2096,2208,2323,2441,2562,2686,2813,2943,3076,3212,3351,3493,3637,3780,3917,4048,4173,4292,4405,4512,4613,4711,4806,4898,4987,5075,5160,5242,5321,5397,5470,5540,5607,5548,5406,5252,5086,4908,4718,4516,4293,4046,3775,3485,3201,2929,2669,2421,2185,1961,1749,1549,1361,1185,1021,869,729,601,485,381,289,331,375,421,470,522,577,635,696,760,827,897,970,1046,1125,1207,1292,1380,1471,1565,1662,1762,1865,1971,2080,2192,2307,2425,2546,2670,2797,2927,3060,3196,3335,3477,3621,3764,3901,4032,4157,4276,4389,4254,3997,3716,3427,3149,2882,3012,3145,3281,3420,3562,3706,3848,3726,3434,3578,3723,3865,4001,4131,4255,4373,4485,4591,4692,4790,4885,4977,5066,5152,5235,5315,5392,5466,5537,5605,5670,5732,5791,5847,5900,5950,5997,6041,6082,6120,6155,6187,6216,6242,6265,6285,6302,6316,6327,6335,6340,6326,6299,6260,6209,6146,6071,5984,5885,5773,5649,5513,5365,5205,5033,4849,4653,4442,4207,3948,3665,3376,3098,2832,2578,2336,2106,1888,1682,1488,1306,1136,978,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,880,953,1029,1108,1190,1275,1363,1454,1548,1645,1745,1848,1954,2063,2175,2290,2408,2529,2653,2780,2910,3043,3179,3318,3460,3604,3747,3886,4019,4146,4267,4382,4491,4594,4694,4602,4391,4156,3897,3614,3325,3047,2781,2527,2285,2055,1837,1631,1437,1255,1085,927,1002,1079,1158,1240,1325,1413,1504,1598,1695,1795,1898,2004,2113,2225,2340,2458,2579,2703,2830,2960,3093,3229,3368,3510,3654,3797,3934,4065,4190,4309,4422,4529,4630,4728,4823,4915,5004,5090,5173,5253,5330,5404,5314,5149,4972,4783,4582,4360,4114,3844,3554,3270,2998,2738,2490,2254,2030,1818,1617,1714,1814,1917,2023,2132,2244,2359,2477,2598,2722,2849,2979,3112,3248,3387,3529,3673,3816,3953,4084,4209,4328,4441,4548,4649,4747,4842,4934,5023,5109,5192,5272,5349,5423,5494,5562,5627,5689,5748,5804,5857,5909,5958,6004,6047,6087,6124,6158,6189,6166,6096,6014,5920,5814,5696,5566,5424,5270,5104,4926,4736,4534,4311,4064,3793,3503,3219,2947,2687,2439,2203,1979,1767,1567,1379,1203,1039,887,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,934,1010,1089,1171,1256,1344,1435,1529,1626,1726,1829,1935,2044,2156,2271,2389,2510,2634,2761,2891,3024,3160,3299,3441,3585,3729,3743,3452,3168,2896,2636,2388,2152,1928,2036,2146,2258,2373,2491,2612,2736,2863,2993,3126,3262,3401,3543,3687,3830,3967,4098,4223,4342,4455,4562,4663,4761,4856,4948,5037,5123,5206,5286,5363,5437,5508,5576,5641,5703,5762,5818,5871,5921,5968,6012,6053,5996,5897,5786,5663,5528,5381,5222,5051,4867,4671,4460,4225,3966,3683,3394,3116,2850,2596,2354,2124,1906,1700,1506,1324,1154,996,850,715,779,846,916,989,1065,1144,1226,1311,1399,1490,1584,1681,1781,1884,1990,2099,2211,2326,2444,2565,2689,2816,2946,3079,3215,3354,3496,3640,3783,3920,4051,4176,4295,4408,4515,4616,4714,4809,4901,4990,5076,5161,5243,5322,5398,5471,5541,5608,5672,5616,5480,5332,5172,5000,4816,4620,4409,4174,3915,3632,3343,3065,2799,2545,2303,2073,1855,1649,1455,1273,1103,945,799,665,543,433,335,249,288,329,372,418,467,519,574,632,693,757,824,894,967,1043,1122,1204,1289,1377,1468,1562,1659,1759,1862,1968,2077,2189,2304,2422,2543,2667,2794,2924,3057,3193,3332,3474,3618,3761,3898,4029,4154,4273,4130,3861,3572,3288,3015,3148,3284,3423,3565,3709,3851,3987,3868,3581,3292,3433,3576,3720,3862,3998,4128,4252,4370,4482,4589,4690,4788,4883,4975,5064,5150,5233,5313,5390,5464,5535,5603,5668,5730,5789,5845,5898,5948,5995,6039,6080,6118,6153,6185,6214,6240,6263,6283,6300,6314,6325,6333,6315,6282,6237,6180,6111,6030,5937,5832,5714,5584,5442,5288,5122,4944,4754,4552,4329,4082,3811,3521,3237,2965,2705,2457,2221,1997,1785,1585,1397,1221,1057,905,765,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,746,813,883,956,1032,1111,1193,1278,1366,1457,1551,1648,1748,1851,1957,2066,2178,2293,2411,2532,2656,2783,2913,3046,3182,3321,3463,3607,3750,3887,4020,4147,4268,4383,4492,4595,4695,4792,4703,4501,4278,4031,3760,3470,3186,2914,2654,2406,2170,1946,1734,1534,1346,1170,1006,854,926,1000,1076,1155,1237,1322,1410,1501,1595,1692,1792,1895,2001,2110,2222,2337,2455,2576,2700,2827,2957,3090,3226,3365,3507,3651,3794,3931,4062,4187,4306,4419,4526,4627,4725,4820,4912,5001,5087,5170,5250,5327,5234,5063,4880,4685,4475,4241,3983,3701,3412,3134,2868,2614,2372,2142,1924,1717,1817,1920,2026,2135,2247,2362,2480,2601,2725,2852,2982,3115,3251,3390,3532,3676,3819,3956,4087,4212,4331,4444,4551,4652,4750,4845,4937,5026,5112,5195,5275,5352,5426,5497,5565,5630,5692,5751,5807,5860,5910,5959,6005,6048,6088,6125,6159,6190,6218,6198,6134,6058,5970,5870,5758,5634,5498,5350,5190,5018,4834,4638,4427,4192,3933,3650,3361,3083,2817,2563,2321,2091,1873,1667,1473,1291,1121,963,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,858,931,1007,1086,1168,1253,1341,1432,1526,1623,1723,1826,1932,2041,2153,2268,2386,2507,2631,2758,2888,3021,3157,3296,3438,3582,3599,3310,3032,2766,2512,2270,2040,1822,1927,2034,2143,2255,2370,2488,2609,2733,2860,2990,3123,3259,3398,3540,3684,3827,3964,4095,4220,4339,4452,4559,4660,4758,4853,4945,5034,5120,5203,5283,5360,5434,5505,5573,5638,5700,5759,5815,5868,5918,5965,6009,5949,5844,5727,5598,5457,5304,5139,4962,4772,4570,4347,4100,3829,3539,3255,2983,2723,2475,2239,2015,1803,1603,1415,1239,1075,923,782,849,919,992,1068,1147,1229,1314,1402,1493,1587,1684,1784,1887,1993,2102,2214,2329,2447,2568,2692,2819,2949,3082,3218,3357,3499,3643,3786,3923,4054,4179,4298,4411,4518,4619,4717,4812,4904,4993,5079,5162,5244,5323,5399,5472,5542,5609,5673,5734,5681,5551,5409,5255,5089,4911,4721,4519,4296,4049,3778,3488,3204,2932,2672,2424,2188,1964,1752,1552,1364,1188,1024,872,732,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,754,821,891,964,1040,1119,1201,1286,1374,1465,1559,1656,1756,1859,1965,2074,2186,2301,2419,2540,2664,2791,2921,3054,3190,3329,3471,3615,3758,3895,4026,4151,4000,3719,3430,3151,3287,3426,3568,3712,3854,3990,4120,4004,3728,3437,3153,3291,3431,3573,3717,3859,3995,4125,4249,4367,4480,4587,4688,4786,4881,4973,5062,5148,5231,5311,5388,5462,5533,5601,5666,5728,5787,5843,5896,5946,5993,6037,6078,6116,6151,6183,6212,6238,6261,6281,6298,6312,6323,6301,6262,6211,6148,6073,5986,5887,5776,5652,5516,5368,5208,5036,4852,4656,4445,4210,3951,3668,3379,3101,2835,2581,2339,2109,1891,1685,1491,1309,1139,981,835,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,749,816,886,959,1035,1114,1196,1281,1369,1460,1554,1651,1751,1854,1960,2069,2181,2296,2414,2535,2659,2786,2916,3049,3185,3324,3466,3610,3753,3890,4021,4148,4269,4384,4493,4596,4696,4793,4887,4801,4605,4394,4159,3900,3617,3328,3050,2784,2530,2288,2058,1840,1634,1440,1258,1088,930,784,853,924,997,1073,1152,1234,1319,1407,1498,1592,1689,1789,1892,1998,2107,2219,2334,2452,2573,2697,2824,2954,3087,3223,3362,3504,3648,3791,3928,4059,4184,4303,4416,4523,4624,4722,4817,4909,4998,5084,5167,5247,5151,4974,4785,4584,4362,4116,3846,3557,3273,3001,2741,2493,2257,2033,1820,1923,2029,2138,2250,2365,2483,2604,2728,2855,2985,3118,3254,3393,3535,3679,3822,3959,4090,4215,4334,4447,4554,4655,4753,4848,4940,5029,5115,5198,5278,5355,5429,5500,5568,5633,5695,5754,5810,5863,5913,5960,6006,6049,6089,6126,6160,6191,6219,6244,6227,6169,6099,6017,5923,5817,5699,5569,5427,5273,5107,4929,4739,4537,4314,4067,3796,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)
Polycyclic presentation:
3,6343|b^a=b^5785
Conjugacy classes & character table
2117 conjugacy classes. The character table is too large to display.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19029, 1);
Finite Group Databasebeta