Finite Group DatabasebetaFinite Group Databasebeta| order | 19731 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C6577⋊C3 |
| Fitting, socle, derived, Sylow | C6577 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C6577 | C3 |
| LCS | G ▷ C6577 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6577 |
| transitive degree | μt(G) = 6577 |
A minimal faithful permutation representation — 2 generators on 6577 points:
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2216,2331,2449,2570,2694,2821,2951,3084,3220,3359,3501,3645,3789,3929,4063,4191,4313,4429,4540,4647,4751,4852,4950,5045,5137,5226,5312,5386,5209,5020,4819,4606,4377,4125,3848,3557,3273,3001,2741,2493,2257,2033,1820,1923,2029,2138,2250,2365,2483,2604,2728,2855,2985,3118,3254,3393,3535,3679,3823,3963,4097,4225,4347,4463,4574,4681,4785,4886,4984,5079,5171,5260,5346,5429,5509,5586,5660,5731,5799,5864,5926,5985,6041,6094,6144,6192,6238,6281,6321,6358,6392,6423,6430,6365,6289,6201,6101,5989,5865,5729,5581,5421,5249,5065,4869,4661,4440,4199,3934,3647,3358,3080,2814,2560,2318,2088,1870,1664,1470,1288,1118,960,814,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,727,794,864,937,1013,1092,1174,1259,1347,1438,1532,1629,1729,1832,1938,2047,2159,2274,2392,2513,2637,2764,2894,3027,3163,3302,3444,3589,3734,3876,4012,3739,3446,3162,2890,2630,2382,2502,2624,2748,2875,3005,3138,3274,3413,3555,3699,3843,3983,4117,4245,4367,4483,4594,4701,4805,4906,5004,5099,5191,5280,5366,5449,5529,5606,5680,5751,5819,5884,5946,6005,6061,6114,6164,6211,6255,6296,6334,6369,6401,6421,6352,6271,6178,6073,5956,5827,5685,5531,5365,5187,4997,4795,4581,4351,4098,3821,3530,3246,2974,2714,2466,2230,2006,1794,1594,1406,1230,1066,914,774,646,530,425,474,526,581,639,700,764,831,901,974,1050,1129,1211,1296,1384,1475,1569,1666,1766,1869,1975,2084,2196,2311,2429,2550,2674,2801,2931,3064,3200,3339,3481,3625,3769,3909,4043,4171,4293,4409,4520,4627,4731,4833,4933,5030,5124,5215,5303,5388,5395,5222,5038,4842,4634,4413,4172,3907,3620,3331,3053,2787,2533,2291,2061,1843,1637,1443,1261,1091,933,787,653,716,781,848,918,991,1067,1146,1228,1313,1401,1492,1586,1683,1783,1886,1992,2101,2213,2328,2446,2567,2691,2818,2948,3081,3217,3356,3498,3642,3786,3926,4060,4188,4310,4426,4537,4644,4748,4849,4947,5042,5134,5223,5300,5117,4922,4715,4495,4255,3991,3704,3415,3137,2871,2617,2375,2145,1926,2032,2141,2253,2368,2486,2607,2731,2858,2988,3121,3257,3396,3538,3682,3826,3966,4100,4228,4350,4466,4577,4684,4788,4889,4987,5082,5174,5263,5349,5432,5512,5589,5663,5734,5802,5867,5929,5988,6044,6097,6147,6194,6239,6282,6322,6359,6393,6424,6452,6459,6400,6330,6248,6154,6048,5930,5800,5658,5504,5338,5160,4970,4768,4554,4324,4071,3794,3503,3219,2947,2687,2439,2203,1979,1767,1567,1379,1203,1039,887,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,934,1010,1089,1171,1256,1344,1435,1529,1626,1726,1829,1935,2044,2156,2271,2389,2510,2634,2761,2891,3024,3160,3299,3441,3586,3731,3873,3594,3304,3026,2760,2506,2264,2381,2500,2621,2745,2872,3002,3135,3271,3410,3552,3696,3840,3980,4114,4242,4364,4480,4591,4698,4802,4903,5001,5096,5188,5277,5363,5446,5526,5603,5677,5748,5816,5881,5943,6002,6058,6111,6161,6208,6252,6293,6331,6366,6389,6314,6227,6128,6017,5894,5759,5611,5451,5279,5095,4899,4691,4470,4229,3964,3677,3388,3110,2844,2590,2348,2118,1900,1694,1500,1318,1148,990,844,710,588,477,529,584,642,703,767,834,904,977,1053,1132,1214,1299,1387,1478,1572,1669,1769,1872,1978,2087,2199,2314,2432,2553,2677,2804,2934,3067,3203,3342,3484,3628,3772,3912,4046,4174,4296,4412,4523,4630,4734,4835,4934,5031,5125,5216,5304,5389,5471,5478,5311,5133,4943,4741,4527,4297,4044,3767,3476,3192,2920,2660,2412,2176,1952,1740,1540,1352,1176,1012,860,720,592,652,714,778,845,915,988,1064,1143,1225,1310,1398,1489,1583,1680,1780,1883,1989,2098,2210,2325,2443,2564,2688,2815,2945,3078,3214,3353,3495,3639,3783,3923,4057,4185,4307,4423,4534,4641,4745,4846,4944,5039,5131,5211,5022,4821,4608,4379,4127,3851,3560,3276,3004,2744,2496,2260,2035,2144,2256,2371,2489,2610,2734,2861,2991,3124,3260,3399,3541,3685,3829,3969,4103,4231,4353,4469,4580,4687,4791,4892,4990,5085,5177,5266,5352,5435,5515,5592,5666,5737,5805,5870,5932,5991,6047,6100,6150,6197,6241,6283,6323,6360,6394,6425,6453,6478,6485,6432,6368,6292,6204,6104,5992,5868,5732,5584,5424,5252,5068,4872,4664,4443,4202,3937,3650,3361,3083,2817,2563,2321,2091,1873,1667,1473,1291,1121,963,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,858,931,1007,1086,1168,1253,1341,1432,1526,1623,1723,1826,1932,2041,2153,2268,2386,2507,2631,2758,2888,3021,3157,3296,3438,3583,3728,3449,3165,2893,2633,2385,2149,2263,2379,2497,2618,2742,2869,2999,3132,3268,3407,3549,3693,3837,3977,4111,4239,4361,4477,4588,4695,4799,4900,4998,5093,5185,5274,5360,5443,5523,5600,5674,5745,5813,5878,5940,5999,6055,6108,6158,6205,6249,6290,6328,6354,6273,6180,6075,5958,5829,5688,5534,5368,5190,5000,4798,4584,4354,4101,3824,3533,3249,2977,2717,2469,2233,2009,1797,1597,1409,1233,1069,917,777,649,532,587,645,706,770,837,907,980,1056,1135,1217,1302,1390,1481,1575,1672,1772,1875,1981,2090,2202,2317,2435,2556,2680,2807,2937,3070,3206,3345,3487,3631,3775,3915,4049,4177,4299,4415,4526,4633,4737,4838,4936,5032,5126,5217,5305,5390,5472,5551,5558,5397,5225,5041,4845,4637,4416,4175,3910,3623,3334,3056,2790,2536,2294,2064,1846,1640,1446,1264,1094,936,790,656,534,591,650,711,775,842,912,985,1061,1140,1222,1307,1395,1486,1580,1677,1777,1880,1986,2095,2207,2322,2440,2561,2685,2812,2942,3075,3211,3350,3492,3636,3780,3920,4054,4182,4304,4420,4531,4638,4742,4843,4941,5036,5119,4924,4717,4497,4257,3993,3707,3418,3140,2874,2620,2378,2147,2259,2374,2492,2613,2737,2864,2994,3127,3263,3402,3544,3688,3832,3972,4106,4234,4356,4472,4583,4690,4794,4895,4993,5088,5180,5269,5355,5438,5518,5595,5669,5740,5808,5873,5935,5994,6050,6103,6153,6200,6244,6285,6324,6361,6395,6426,6454,6479,6501,6508,6461,6403,6333,6251,6157,6051,5933,5803,5661,5507,5341,5163,4973,4771,4557,4327,4074,3797,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)Polycyclic presentation:
3,6577|b^a=b^62232195 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19731, 1);