Finite Group DatabasebetaFinite Group Databasebeta| order | 19911 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C6637⋊C3 |
| Fitting, socle, derived, Sylow | C6637 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C6637 | C3 |
| LCS | G ▷ C6637 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6637 |
| transitive degree | μt(G) = 6637 |
A minimal faithful permutation representation — 2 generators on 6637 points:
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7,2161,1937,1725,1525,1337,1427,1519,1613,1710,1810,1913,2019,2128,2240,2355,2473,2594,2718,2845,2975,3108,3244,3383,3525,3670,3818,3968,4118,4264,4404,4538,4666,4788,4904,5014,5118,5216,5308,5395,5478,5558,5635,5709,5780,5848,5899,5758,5605,5440,5259,5055,4827,4576,4301,4003,3702,3412,3134,2868,2614,2372,2142,1924,1717,1817,1920,2026,2135,2247,2362,2480,2601,2725,2852,2982,3115,3251,3390,3532,3677,3825,3975,4125,4271,4411,4545,4673,4795,4911,5021,5125,5223,5315,5402,5485,5565,5642,5716,5787,5855,5920,5982,6041,6098,6153,6205,6254,6300,6343,6383,6420,6454,6485,6513,6538,6546,6494,6431,6356,6269,6169,6057,5933,5797,5649,5489,5316,5123,4906,4665,4400,4111,3808,3512,3228,2956,2696,2448,2212,1988,1776,1576,1388,1212,1048,896,756,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,728,795,865,938,1014,1093,1175,1260,1348,1439,1533,1630,1730,1833,1939,2048,2160,2275,2393,2514,2638,2765,2895,3028,3164,3303,3445,3590,3738,3889,4040,4188,4330,4466,4493,4215,3916,3615,3325,3047,2781,2527,2285,2055,1837,1631,1437,1255,1085,927,1002,1079,1158,1240,1325,1413,1504,1598,1695,1795,1898,2004,2113,2225,2340,2458,2579,2703,2830,2960,3093,3229,3368,3510,3655,3803,3953,4103,4249,4389,4523,4651,4773,4889,4999,5103,5201,5293,5380,5463,5529,5357,5166,4951,4712,4449,4162,3859,3563,3279,3007,2747,2499,2262,2377,2495,2616,2740,2867,2997,3130,3266,3405,3547,3692,3840,3990,4140,4286,4426,4560,4688,4810,4926,5036,5140,5238,5330,5417,5500,5580,5657,5731,5802,5870,5935,5997,6056,6112,6165,6215,6262,6306,6348,6388,6425,6459,6490,6518,6543,6565,6584,6600,6613,6623,6626,6604,6571,6526,6469,6400,6319,6226,6121,6004,5874,5732,5578,5412,5230,5025,4796,4543,4266,3967,3666,3376,3098,2832,2578,2336,2106,1888,1682,1488,1306,1136,978,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,880,953,1029,1108,1190,1275,1363,1454,1548,1645,1745,1848,1954,2063,2175,2290,2408,2529,2653,2780,2910,3043,3179,3318,3460,3605,3753,3904,4055,4203,4345,4481,4611,4735,4853,4965,5071,5088,4870,4629,4364,4075,3772,3476,3192,2920,2660,2412,2176,1952,1740,1540,1352,1176,1012,860,720,592,652,714,778,845,915,988,1064,1143,1225,1310,1398,1489,1583,1680,1780,1883,1989,2098,2210,2325,2443,2564,2688,2815,2945,3078,3214,3353,3495,3640,3788,3938,4088,4234,4374,4508,4636,4758,4874,4984,5069,4842,4591,4316,4018,3717,3427,3149,2882,3012,3145,3281,3420,3562,3707,3855,4005,4155,4300,4439,4183,3881,3579,3726,3875,4025,4175,4320,4459,4592,4719,4840,4955,5064,5167,5264,5356,5443,5526,5606,5683,5757,5828,5896,5961,6023,6082,6138,6191,6241,6288,6332,6373,6411,6446,6478,6507,6533,6556,6576,6593,6607,6618,6621,6594,6555,6504,6441,6366,6279,6180,6069,5946,5811,5664,5504,5331,5138,4921,4680,4415,4126,3823,3527,3243,2971,2711,2463,2227,2003,1791,1591,1403,1227,1063,911,771,643,527,423,330,373,419,468,520,575,633,694,758,825,895,968,1044,1123,1205,1290,1378,1469,1563,1660,1760,1863,1969,2078,2190,2305,2423,2544,2668,2795,2925,3058,3194,3333,3475,3620,3768,3918,4068,4214,4355,4491,4621,4745,4863,4975,5081,5181,5275,5364,5449,5531,5543,5376,5194,4989,4760,4507,4230,3931,3630,3340,3062,2796,2542,2300,2070,1852,1646,1452,1270,1100,942,796,662,540,430,332,377,424,473,525,580,638,699,763,830,900,973,1049,1128,1210,1295,1383,1474,1568,1665,1765,1868,1974,2083,2195,2310,2428,2549,2673,2800,2930,3063,3199,3338,3480,3625,3773,3923,4073,4219,4359,4464,4177,3874,3577,3722,3870,4020,4170,4315,4454,4587,4714,4835,4950,5059,4848,4603,4334,4041,3736,3440,3156,2884,3016,3150,3286,3425,3567,3712,3860,4010,4160,4305,4444,4577,4704,4826,4942,5052,5156,5254,5346,5433,5516,5596,5673,5747,5818,5886,5951,6013,6072,6128,6181,6231,6278,6322,6363,6401,6436,6468,6497,6523,6536,6479,6410,6329,6236,6131,6014,5885,5744,5591,5426,5245,5040,4811,4558,4281,3982,3681,3391,3113,2847,2593,2351,2121,1903,1697,1503,1321,1151,993,847,713,590,648,709,773,840,910,983,1059,1138,1220,1305,1393,1484,1578,1675,1775,1878,1984,2093,2205,2320,2438,2559,2683,2810,2940,3073,3209,3348,3490,3635,3783,3933,4083,4229,4369,4503,4631,4753,4869,4980,5086,5186,5280,5369,5454,5536,5615,5691,5764,5834,5901,5913,5776,5628,5468,5295,5102,4885,4644,4379,4090,3787,3491,3207,2935,2675,2427,2191,1967,1755,1555,1367,1191,1027,875,735,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,748,815,885,958,1034,1113,1195,1280,1368,1459,1553,1650,1750,1853,1959,2068,2180,2295,2413,2534,2658,2785,2915,3048,3184,3323,3465,3610,3758,3909,4060,4208,4350,4486,4616,4740,4858,4970,5076,5176,4969,4736,4479,4198,3896,3594,3304,3026,2760,2506,2264,2381,2500,2621,2745,2872,3002,3135,3271,3410,3552,3697,3845,3995,4145,4291,4431,4565,4693,4815,4931,5041,5145,5243,5335,5422,5505,5585,5662,5736,5807,5875,5940,6002,6061,6117,6170,6220,6267,6311,6352,6376,6289,6190,6079,5956,5821,5674,5515,5343,5151,4935,4695,4430,4141,3838,3542,3258,2986,2726,2478,2242,2018,1806,1606,1418,1242,1078,925,998,1074,1153,1235,1320,1408,1499,1593,1690,1790,1893,1999,2108,2220,2335,2453,2574,2698,2825,2955,3088,3224,3363,3505,3650,3798,3948,4098,4244,4384,4518,4646,4768,4884,4994,5098,5196,5288,5375,5459,5541,5620,5696,5769,5839,5906,5970,6031,6089,6144,6196,6208,6101,5983,5853,5711,5557,5391,5209,5004,4775,4522,4245,3946,3645,3355,3077,2811,2557,2315,2085,1867,1661,1467,1285,1115,957,811,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,733,800,870,943,1019,1098,1180,1265,1353,1444,1538,1635,1735,1838,1944,2053,2165,2280,2398,2519,2643,2770,2900,3033,3169,3308,3450,3595,3743,3894,4045,4193,4335,4471,4601,4349,4056,3751,3455,3171,2899,2639,2391,2155,1931,1719,1821,1925,2031,2140,2252,2367,2485,2606,2730,2857,2987,3120,3256,3395,3537,3682,3830,3980,4130,4276,4416,4550,4678,4800,4916,5026,5130,5228,5320,5407,5490,5570,5647,5721,5792,5860,5925,5987,6046,6102,6141,6024,5895,5754,5601,5436,5255,5051,4823,4571,4295,3997,3696,3406,3128,2862,2608,2366,2136,1918,1712,1518,1335,1423,1514,1608,1705,1805,1908,2014,2123,2235,2350,2468,2589,2713,2840,2970,3103,3239,3378,3520,3665,3813,3963,4113,4259,4399,4533,4661,4783,4899,5009,5113,5211,5303,5390,5473,5553,5630,5704,5775,5844,5911,5975,6036,6094,6149,6201,6250,6296,6339,6379,6416,6428,6351,6263,6163,6051,5927,5791,5643,5483,5310,5117,4900,4659,4394,4105,3802,3506,3222,2950,2690,2442,2206,1982,1770,1570,1382,1206,1042,890,750,622,506,402,310,230,162,106,62,30,10)Polycyclic presentation:
3,6637|b^a=b^52662215 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19911, 1);