Finite Group DatabasebetaFinite Group Databasebeta| order | 20073 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C6691⋊C3 |
| Fitting, socle, derived, Sylow | C6691 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C6691 | C3 |
| LCS | G ▷ C6691 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6691 |
| transitive degree | μt(G) = 6691 |
A minimal faithful permutation representation — 2 generators on 6691 points:
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,2013,2122,2234,2349,2467,2588,2712,2839,2969,3102,3238,3377,3519,3664,3809,3950,4085,4214,4337,4454,4567,4677,4784,4888,4989,5087,5182,5274,5363,5449,5532,5612,5689,5763,5834,5902,5967,6029,6089,6146,6200,6251,6188,6070,5940,5798,5644,5478,5300,5110,4908,4694,4468,4225,3958,3669,3379,3101,2835,2581,2339,2109,1891,1685,1491,1309,1139,981,835,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,749,816,886,959,1035,1114,1196,1281,1369,1460,1554,1651,1751,1854,1960,2069,2181,2296,2414,2535,2659,2786,2916,3049,3185,3324,3466,3611,3756,3897,4032,4161,4284,4401,4271,4014,3733,3440,3156,2884,3016,3150,3286,3425,3567,3711,3856,3997,4132,4261,4384,4501,4614,4724,4831,4935,5036,5134,5229,5321,5410,5496,5579,5659,5736,5810,5881,5949,6014,6076,6135,6191,6244,6294,6341,6385,6426,6464,6499,6531,6560,6586,6609,6629,6646,6660,6672,6681,6687,6690,6679,6656,6621,6574,6516,6446,6364,6270,6164,6046,5916,5774,5620,5454,5276,5086,4884,4670,4444,4201,3934,3645,3355,3077,2811,2557,2315,2085,1867,1661,1467,1285,1115,957,811,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,733,800,870,943,1019,1098,1180,1265,1353,1444,1538,1635,1735,1838,1944,2053,2165,2280,2398,2519,2643,2770,2900,3033,3169,3308,3450,3595,3740,3882,4018,4148,4272,4129,3850,3557,3273,3001,2741,2493,2257,2033,1820,1923,2029,2138,2250,2365,2483,2604,2728,2855,2985,3118,3254,3393,3535,3680,3825,3966,4101,4230,4353,4470,4583,4693,4800,4904,5005,5103,5198,5290,5379,5465,5548,5628,5705,5779,5850,5918,5983,6045,6104,6160,6213,6263,6310,6250,6143,6024,5893,5750,5596,5430,5252,5062,4860,4646,4420,4177,3910,3621,3331,3053,2787,2533,2291,2061,1843,1637,1443,1261,1091,933,787,653,716,781,848,918,991,1067,1146,1228,1313,1401,1492,1586,1683,1783,1886,1992,2101,2213,2328,2446,2567,2691,2818,2948,3081,3217,3356,3498,3643,3788,3929,4064,4193,4316,4433,4546,4656,4763,4867,4968,5066,5161,5253,5342,5428,5511,5592,5670,5745,5817,5733,5573,5401,5217,5021,4813,4593,4360,4105,3826,3533,3249,2977,2717,2469,2233,2009,1797,1597,1409,1233,1069,917,777,649,532,587,645,706,770,837,907,980,1056,1135,1217,1302,1390,1481,1575,1672,1772,1875,1981,2090,2202,2317,2435,2556,2680,2807,2937,3070,3206,3345,3487,3632,3777,3918,4053,4182,4305,4422,4535,4645,4752,4856,4957,5055,5150,5042,4839,4624,4397,4153,3886,3597,3307,3029,2763,2509,2267,2037,2148,2261,2376,2494,2615,2739,2866,2996,3129,3265,3404,3546,3691,3836,3977,4112,4241,4364,4481,4594,4704,4811,4915,5016,5114,5209,5301,5390,5476,5559,5639,5716,5790,5861,5929,5994,6056,6115,6171,6224,6274,6321,6365,6406,6444,6479,6512,6542,6569,6593,6557,6493,6417,6329,6229,6117,5993,5857,5709,5549,5377,5193,4997,4789,4569,4336,4081,3802,3509,3225,2953,2693,2445,2209,1985,1773,1573,1385,1209,1045,893,753,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,722,789,859,932,1008,1087,1169,1254,1342,1433,1527,1624,1724,1827,1933,2042,2154,2269,2387,2508,2632,2759,2889,3022,3158,3297,3439,3584,3729,3871,4000,3713,3424,3146,2880,2625,2749,2876,3006,3139,3275,3414,3556,3701,3846,3987,4122,4251,4374,4491,4604,4714,4821,4925,5026,5124,5219,5311,5400,5486,5569,5649,5726,5800,5871,5939,6004,6066,6125,6181,6234,6284,6331,6375,6416,6454,6489,6521,6550,6576,6537,6472,6395,6306,6205,6093,5969,5833,5685,5525,5353,5169,4973,4765,4545,4312,4057,3778,3485,3201,2929,2669,2421,2185,1961,1749,1549,1361,1185,1021,869,729,601,485,381,289,331,375,421,470,522,577,635,696,760,827,897,970,1046,1125,1207,1292,1380,1471,1565,1662,1762,1865,1971,2080,2192,2307,2425,2546,2670,2797,2927,3060,3196,3335,3477,3622,3767,3908,4043,4172,4295,4412,4525,4635,4742,4846,4948,5047,5143,5236,5131,4929,4715,4489,4246,3979,3690,3400,3122,2856,2602,2360,2130,1912,1706,1512,1330,1160,1001,1077,1156,1238,1323,1411,1502,1596,1693,1793,1896,2002,2111,2223,2338,2456,2577,2701,2828,2958,3091,3227,3366,3508,3653,3798,3939,4074,4203,4326,4443,4556,4666,4773,4877,4978,5076,5171,5263,5352,5438,5521,5601,5678,5752,5665,5504,5331,5146,4949,4741,4521,4288,4033,3754,3461,3177,2905,2645,2397,2161,1937,1725,1525,1337,1427,1519,1613,1710,1810,1913,2019,2128,2240,2355,2473,2594,2718,2845,2975,3108,3244,3383,3525,3670,3815,3956,4091,4220,4343,4460,4573,4683,4790,4894,4995,5093,5188,5280,5369,5455,5538,5618,5695,5769,5840,5908,5973,6035,6094,6150,6204,6255,6303,6348,6291,6185,6067,5937,5795,5641,5475,5297,5107,4905,4691,4465,4222,3955,3666,3376,3098,2832,2578,2336,2106,1888,1682,1488,1306,1136,978,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,880,953,1029,1108,1190,1275,1363,1454,1548,1645,1745,1848,1954,2063,2175,2290,2408,2529,2653,2780,2910,3043,3179,3318,3460,3605,3750,3891,4026,4155,4011,3730,3437,3153,3291,3431,3573,3717,3861,3869,3577,3721,3865,4004,4137,4265,4388,4505,4618,4728,4835,4939,5040,5138,5233,5325,5414,5500,5583,5663,5740,5814,5885,5953,6018,6080,6139,6195,6248,6298,6345,6389,6430,6468,6503,6535,6564,6590,6613,6633,6650,6664,6675,6683,6688,6677,6654,6619,6572,6513,6443,6361,6267,6161,6043,5913,5771,5617,5451,5273,5083,4881,4667,4441,4198,3931,3642,3352,3074,2808,2554,2312,2082,1864,1658,1464,1282,1112,954,808,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,739,806,876,949,1025,1104,1186,1271,1359,1450,1544,1641,1741,1844,1950,2059,2171,2286,2404,2525,2649,2776,2906,3039,3175,3314,3456,3601,3746,3887,4022,4152,4276,4394,4508,4381,4126,3847,3554,3270,2998,2738,2490,2254,2030,1818,1617,1714,1814,1917,2023,2132,2244,2359,2477,2598,2722,2849,2979,3112,3248,3387,3529,3674,3819,3960,4095,4224,4347,4464,4577,4687,4794,4898,4999,5097,5192,5284,5373,5459,5542,5622,5699,5773,5844,5912,5977,6039,6098,6154,6207,6141,6022,5891,5748,5593,5427,5249,5059,4857,4643,4417,4174,3907,3618,3328,3050,2784,2530,2288,2058,1840,1634,1440,1258,1088,930,784,853,924,997,1073,1152,1234,1319,1407,1498,1592,1689,1789,1892,1998,2107,2219,2334,2452,2573,2697,2824,2954,3087,3223,3362,3504,3649,3794,3935,4070,4199,4322,4439,4552,4662,4769,4873,4974,5072,5167,5259,5348,5434,5517,5597,5674,5749,5821,5890,5956,5878,5730,5570,5398,5214,5018,4810,4590,4357,4102,3823,3530,3246,2974,2714,2466,2230,2006,1794,1594,1406,1230,1066,914,774,646,530,425,474,526,581,639,700,764,831,901,974,1050,1129,1211,1296,1384,1475,1569,1666,1766,1869,1975,2084,2196,2311,2429,2550,2674,2801,2931,3064,3200,3339,3481,3626,3771,3912,4047,4176,4299,4416,4529,4639,4746,4850,4951,4837,4622,4395,4151,3883,3594,3304,3026,2760,2506,2264,2381,2500,2621,2745,2872,3002,3135,3271,3410,3552,3697,3842,3983,4118,4247,4370,4487,4600,4710,4817,4921,5022,5120,5215,5307,5396,5482,5565,5645,5722,5796,5867,5935,6000,6062,6121,6177,6230,6280,6327,6371,6412,6450,6485,6517,6546,6573,6597,6618,6636,6606,6554,6490,6414,6326,6226,6114,5990,5854,5706,5546,5374,5190,4994,4786,4566,4333,4078,3799,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)Polycyclic presentation:
3,6691|b^a=b^37722233 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20073, 1);