Finite Group DatabasebetaFinite Group Databasebeta| order | 20199 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C6733⋊C3 |
| Fitting, socle, derived, Sylow | C6733 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C6733 | C3 |
| LCS | G ▷ C6733 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6733 |
| transitive degree | μt(G) = 6733 |
A minimal faithful permutation representation — 2 generators on 6733 points:
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41,6398,6449,6419,6305,6168,6007,5822,5613,5380,5123,4842,4537,4216,3902,3600,3310,3032,2766,2512,2270,2040,1822,1927,2034,2143,2255,2370,2488,2609,2733,2860,2990,3123,3259,3398,3540,3685,3833,3984,4138,4295,4454,4611,4762,4507,4187,4344,4503,4660,4811,4956,5095,5228,5355,5476,5591,5700,5803,5900,5991,6076,6155,6228,6295,6356,6411,6460,6505,6544,6577,6606,6632,6655,6675,6692,6706,6717,6725,6719,6695,6660,6613,6554,6478,6378,6254,6106,5934,5737,5516,5271,5002,4709,4393,4074,3766,3470,3186,2914,2654,2406,2170,1946,1734,1534,1346,1170,1006,854,926,1000,1076,1155,1237,1322,1410,1501,1595,1692,1792,1895,2001,2110,2222,2337,2455,2576,2700,2827,2957,3090,3226,3365,3507,3652,3800,3951,4105,4262,4421,4579,4732,4879,5021,5157,5287,5411,5529,5641,5747,5847,5941,6029,6111,6187,6257,6321,6379,6431,6477,6517,6551,6581,6596,6530,6442,6330,6194,6034,5850,5642,5410,5154,4874,4570,4249,3935,3633,3343,3065,2799,2545,2303,2073,1855,1649,1455,1273,1103,945,799,665,543,433,335,249,288,329,372,418,467,519,574,632,693,757,824,894,967,1043,1122,1204,1289,1377,1468,1562,1659,1759,1862,1968,2077,2189,2304,2422,2543,2667,2794,2924,3057,3193,3332,3474,3619,3767,3918,4072,4229,4388,4547,4701,4849,4991,5127,5257,5381,5499,5611,5717,5817,5911,5956,5760,5541,5298,5031,4740,4426,4107,3799,3503,3219,2947,2687,2439,2203,1979,1767,1567,1379,1203,1039,887,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,934,1010,1089,1171,1256,1344,1435,1529,1626,1726,1829,1935,2044,2156,2271,2389,2510,2634,2761,2891,3024,3160,3299,3441,3586,3734,3885,4039,4196,4355,4514,4602,4282,3968,3666,3376,3098,2832,2578,2336,2106,1888,1682,1488,1306,1136,978,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,880,953,1029,1108,1190,1275,1363,1454,1548,1645,1745,1848,1954,2063,2175,2290,2408,2529,2653,2780,2910,3043,3179,3318,3460,3605,3753,3904,4058,4215,4374,4533,4687,4835,4977,5113,5243,5367,5485,5598,5559,5320,5057,4770,4459,4140,3832,3536,3252,2980,2720,2472,2236,2012,1800,1600,1412,1236,1072,920,780,651,712,776,843,913,986,1062,1141,1223,1308,1396,1487,1581,1678,1778,1881,1987,2096,2208,2323,2441,2562,2686,2813,2943,3076,3212,3351,3493,3638,3786,3937,4091,4248,4407,4566,4720,4868,5010,5146,5276,5400,5518,5630,5736,5836,5930,6018,6100,6176,6246,6310,6368,6420,6467,6461,6354,6223,6068,5889,5686,5459,5208,4933,4634,4315,4001,3699,3409,3131,2865,2611,2369,2139,1921,1715,1520,1614,1711,1811,1914,2020,2129,2241,2356,2474,2595,2719,2846,2976,3109,3245,3384,3526,3671,3819,3970,4124,4281,4440,4598,4751,4898,5039,5174,5303,5426,5543,5654,5599,5365,5108,4827,4522,4201,3887,3585,3295,3017,3152,3289,3428,3570,3715,3863,4014,4168,4325,4484,4641,4792,4937,5076,5209,5336,5457,5572,5681,5784,5881,5972,5807,5592,5353,5090,4803,4492,4173,3865,3569,3285,3013,2752,2879,3009,3142,3278,3417,3559,3704,3852,4003,4157,4314,4473,4630,4781,4926,5065,5198,5325,5446,5561,5670,5775,5874,5967,6054,6135,6210,6279,6342,6399,6450,6495,6468,6366,6241,6092,5919,5722,5501,5256,4987,4694,4378,4059,3751,3455,3171,2899,2639,2391,2155,1931,1719,1821,1925,2031,2140,2252,2367,2485,2606,2730,2857,2987,3120,3256,3395,3537,3682,3830,3981,4135,4292,4451,4608,4347,4506,4663,4814,4959,5098,5231,5358,5479,5594,5703,5806,5903,5994,6079,6158,6231,6298,6359,6414,6463,6506,6545,6578,6607,6633,6656,6676,6693,6707,6718,6726,6731,6727,6709,6680,6639,6586,6520,6432,6320,6184,6024,5840,5631,5398,5141,4860,4555,4234,3920,3618,3328,3050,2784,2530,2288,2058,1840,1634,1440,1258,1088,930,784,853,924,997,1073,1152,1234,1319,1407,1498,1592,1689,1789,1892,1998,2107,2219,2334,2452,2573,2697,2824,2954,3087,3223,3362,3504,3649,3797,3948,4102,4259,4418,4576,4729,4877,5019,5155,5285,5409,5527,5639,5745,5845,5939,6027,6109,6185,6255,6319,6377,6429,6475,6515,6549,6566,6490,6390,6266,6118,5946,5750,5530,5286,5018,4726,4411,4092,3784,3488,3204,2932,2672,2424,2188,1964,1752,1552,1364,1188,1024,872,732,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,754,821,891,964,1040,1119,1201,1286,1374,1465,1559,1656,1756,1859,1965,2074,2186,2301,2419,2540,2664,2791,2921,3054,3190,3329,3471,3616,3764,3915,4069,4226,4385,4544,4698,4846,4988,5124,5254,5378,5496,5608,5714,5814,5862,5655,5424,5169,4890,4587,4267,3953,3651,3361,3083,2817,2563,2321,2091,1873,1667,1473,1291,1121,963,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,858,931,1007,1086,1168,1253,1341,1432,1526,1623,1723,1826,1932,2041,2153,2268,2386,2507,2631,2758,2888,3021,3157,3296,3438,3583,3731,3882,4036,4193,4352,4444,4125,3817,3521,3237,2965,2705,2457,2221,1997,1785,1585,1397,1221,1057,905,765,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,746,813,883,956,1032,1111,1193,1278,1366,1457,1551,1648,1748,1851,1957,2066,2178,2293,2411,2532,2656,2783,2913,3046,3182,3321,3463,3608,3756,3907,4061,4218,4377,4536,4690,4838,4980,5116,5246,5370,5488,5600,5707,5671,5444,5193,4918,4619,4300,3986,3684,3394,3116,2850,2596,2354,2124,1906,1700,1506,1324,1154,996,850,715,779,846,916,989,1065,1144,1226,1311,1399,1490,1584,1681,1781,1884,1990,2099,2211,2326,2444,2565,2689,2816,2946,3079,3215,3354,3496,3641,3789,3940,4094,4251,4410,4569,4723,4871,5013,5149,5279,5403,5521,5633,5739,5839,5933,6021,6103,6179,6249,6313,6371,6423,6469,6510,6507,6412,6293,6150,5983,5792,5577,5338,5075,4788,4477,4158,3850,3554,3270,2998,2738,2490,2254,2030,1818,1617,1714,1814,1917,2023,2132,2244,2359,2477,2598,2722,2849,2979,3112,3248,3387,3529,3674,3822,3973,4127,4284,4443,4601,4754,4901,5042,5177,5306,5429,5546,5657,5762,5708,5486,5241,4972,4679,4363,4044,3736,3440,3156,2884,3016,3150,3286,3425,3567,3712,3860,4011,4165,4322,4481,4638,4789,4934,5073,5206,5333,5454,5569,5678,5781,5878,5704,5477,5226,4951,4652,4333,4019,3717,3427,3149,2882,3012,3145,3281,3420,3562,3707,3855,4006,4160,4317,4476,4633,4784,4929,5068,5201,5328,5449,5564,5673,5776,5875,5968,6055,6136,6211,6280,6343,6400,6451,6496,6535,6511,6421,6308,6171,6010,5825,5616,5383,5126,4845,4540,4219,3905,3603,3313,3035,2769,2515,2273,2043,1825,1619,1718,1819,1922,2028,2137,2249,2364,2482,2603,2727,2854,2984,3117,3253,3392,3534,3679,3827,3978,4132,4289,4448,4605,4758,4905,5046,5181,5310,5433,5550,5661,5766,5865,5958,6046,6128,6204,6274,6338,6396,6448,6494,6534,6568,6598,6625,6649,6670,6688,6703,6715,6724,6730,6733,6732,6720,6697,6662,6615,6556,6480,6380,6256,6108,5936,5740,5519,5274,5005,4712,4396,4077,3769,3473,3189,2917,2657,2409,2173,1949,1737,1537,1349,1173,1009,857,717,783,851,921,994,1070,1149,1231,1316,1404,1495,1589,1686,1786,1889,1995,2104,2216,2331,2449,2570,2694,2821,2951,3084,3220,3359,3501,3646,3794,3945,4099,4256,4415,4573,4727,4875,5017,5153,5283,5407,5525,5637,5743,5843,5937,6025,6107,6183,6253,6317,6375,6427,6473,6513,6532,6444,6332,6196,6036,5852,5644,5412,5156,4876,4572,4252,3938,3636,3346,3068,2802,2548,2306,2076,1858,1652,1458,1276,1106,948,802,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,751,818,888,961,1037,1116,1198,1283,1371,1462,1556,1653,1753,1856,1962,2071,2183,2298,2416,2537,2661,2788,2918,3051,3187,3326,3468,3613,3761,3912,4066,4223,4382,4541,4695,4843,4985,5121,5251,5375,5493,5605,5711,5763,5544,5301,5034,4743,4429,4110,3802,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)Polycyclic presentation:
3,6733|b^a=b^61132247 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20199, 1);