Finite Group DatabasebetaFinite Group Databasebeta| order | 32622 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D5437⋊C3 |
| unambiguous semidirect product | (C5437⋊C3)⋊C2 |
| Fitting, socle, derived, Sylow | C5437 ⋊ C2×C3 |
| Hall | C5437⋊C3 ⋊ C2 |
| Hall | D5437 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5437 | C2×C3 |
| LCS | G ▷ C5437 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 5437 |
| transitive degree | μt(G) = 5437 |
A minimal faithful permutation representation — 2 generators on 5437 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,328)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,287,288,236,237,262)(198,294,295,243,244,257)(203,301,302,249,250,252)(208,307,299,254,255,247)(212,312,292,259,260,240)(215,317,285,263,264,233)(226,333,334,277,278,393)(228,331,332,280,276,325)(231,270,330,282,283,326)(238,346,347,289,290,321)(245,353,354,296,297,316)(251,360,361,303,304,311)(256,367,365,308,309,306)(261,372,358,313,314,300)(265,377,351,318,319,293)(268,382,344,322,323,286)(279,398,399,336,337,464)(281,396,397,339,335,390)(284,329,395,341,342,391)(291,411,412,348,349,386)(298,418,419,355,356,381)(305,425,426,362,363,376)(310,432,433,368,369,371)(315,438,430,373,374,366)(320,443,423,378,379,359)(324,448,416,383,384,352)(327,453,409,387,388,345)(338,469,470,401,402,541)(340,467,468,404,400,461)(343,394,466,406,407,462)(350,482,483,413,414,457)(357,489,490,420,421,452)(364,496,497,427,428,447)(370,503,504,434,435,442)(375,510,508,439,440,437)(380,515,501,444,445,431)(385,520,494,449,450,424)(389,525,487,454,455,417)(392,530,480,458,459,410)(403,546,547,472,473,624)(405,544,545,475,471,538)(408,465,543,477,478,539)(415,559,560,484,485,534)(422,566,567,491,492,529)(429,573,574,498,499,524)(436,580,581,505,506,519)(441,587,588,511,512,514)(446,593,585,516,517,509)(451,598,578,521,522,502)(456,603,571,526,527,495)(460,608,564,531,532,488)(463,613,557,535,536,481)(474,629,630,549,550,713)(476,627,628,552,548,621)(479,542,626,554,555,622)(486,642,643,561,562,617)(493,649,650,568,569,612)(500,656,657,575,576,607)(507,663,664,582,583,602)(513,670,671,589,590,597)(518,677,675,594,595,592)(523,682,668,599,600,586)(528,687,661,604,605,579)(533,692,654,609,610,572)(537,697,647,614,615,565)(540,702,640,618,619,558)(551,718,719,632,633,808)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,731,732,644,645,706)(570,738,739,651,652,701)(577,745,746,658,659,696)(584,752,753,665,666,691)(591,759,760,672,673,686)(596,766,767,678,679,681)(601,772,764,683,684,676)(606,777,757,688,689,669)(611,782,750,693,694,662)(616,787,743,698,699,655)(620,792,736,703,704,648)(623,797,729,707,708,641)(634,813,814,721,722,909)(636,811,812,724,720,805)(639,714,810,726,727,806)(646,826,827,733,734,801)(653,833,834,740,741,796)(660,840,841,747,748,791)(667,847,848,754,755,786)(674,854,855,761,762,781)(680,861,862,768,769,776)(685,868,866,773,774,771)(690,873,859,778,779,765)(695,878,852,783,784,758)(700,883,845,788,789,751)(705,888,838,793,794,744)(709,893,831,798,799,737)(712,898,824,802,803,730)(723,914,915,816,817,1016)(725,912,913,819,815,906)(728,809,911,821,822,907)(735,927,928,828,829,902)(742,934,935,835,836,897)(749,941,942,842,843,892)(756,948,949,849,850,887)(763,955,956,856,857,882)(770,962,963,863,864,877)(775,969,970,869,870,872)(780,975,967,874,875,867)(785,980,960,879,880,860)(790,985,953,884,885,853)(795,990,946,889,890,846)(800,995,939,894,895,839)(804,1000,932,899,900,832)(807,1005,925,903,904,825)(818,1021,1022,917,918,1129)(820,1019,1020,920,916,1013)(823,910,1018,922,923,1014)(830,1034,1035,929,930,1009)(837,1041,1042,936,937,1004)(844,1048,1049,943,944,999)(851,1055,1056,950,951,994)(858,1062,1063,957,958,989)(865,1069,1070,964,965,984)(871,1076,1077,971,972,979)(876,1083,1081,976,977,974)(881,1088,1074,981,982,968)(886,1093,1067,986,987,961)(891,1098,1060,991,992,954)(896,1103,1053,996,997,947)(901,1108,1046,1001,1002,940)(905,1113,1039,1006,1007,933)(908,1118,1032,1010,1011,926)(919,1134,1135,1024,1025,1248)(921,1132,1133,1027,1023,1126)(924,1017,1131,1029,1030,1127)(931,1147,1148,1036,1037,1122)(938,1154,1155,1043,1044,1117)(945,1161,1162,1050,1051,1112)(952,1168,1169,1057,1058,1107)(959,1175,1176,1064,1065,1102)(966,1182,1183,1071,1072,1097)(973,1189,1190,1078,1079,1092)(978,1196,1197,1084,1085,1087)(983,1202,1194,1089,1090,1082)(988,1207,1187,1094,1095,1075)(993,1212,1180,1099,1100,1068)(998,1217,1173,1104,1105,1061)(1003,1222,1166,1109,1110,1054)(1008,1227,1159,1114,1115,1047)(1012,1232,1152,1119,1120,1040)(1015,1237,1145,1123,1124,1033)(1026,1253,1254,1137,1138,1373)(1028,1251,1252,1140,1136,1245)(1031,1130,1250,1142,1143,1246)(1038,1266,1267,1149,1150,1241)(1045,1273,1274,1156,1157,1236)(1052,1280,1281,1163,1164,12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45,5392,5405,5368,5313,5240,5149,5040,4913,4770,4584,4371,4149,3933,3723,3519,3321,3129,2943,2762,2430,2097,2256,2270,2277,2284,2291,2298,2305,2312,2319,2326,2333,2340,2347,2352,2357,2362,2367,2372,2377,2382,2387,2392,2397,2402,2407,2411,2414,2416,2586,2761,2941,3127,3319,3517,3721,3931,4147,4369,4582,4422,4636,4805,4801,4797,4793,4789,4785,4781,4777,4773,4769,4765,4760,4755,4750,4745,4740,4735,4730,4724,4717,4710,4703,4696,4689,4685,4424,4639,4645,4650,4655,4660,4663,4658,4653,4648,4643,4637,4421,4635,4692,4699,4706,4713,4720,4727,4732,4737,4742,4747,4752,4757,4762,4767,4771,4775,4779,4783,4787,4791,4795,4799,4803,4807,4810,4812,4682,4487,4264,4036,3814,3598,3388,3184,2986,2794,2608,2420,2094,2251,2405,2400,2395,2390,2385,2380,2375,2370,2365,2360,2355,2350,2344,2337,2330,2323,2316,2309,2302,2295,2288,2281,2274,2267,2252,2093,2250,2415,2585,2760,2940,3126,3318,3516,3720,3930,4146,4368,4581,4768,4911,5038,5147,5238,5311,5366,5389,5342,5277,5194,5093,4974,4839,4694,4508,4296,4075,3860,3651,3448,3251,3060,2875,2692,2512,2338,2170,2008,1852,1702,1558,1420,1288,1162,1042,928,812,630,715,797,792,787,782,777,772,766,759,752,745,738,731,716,629,714,807,905,1008,1116,1230,1350,1476,1608,1746,1890,2040,2196,2358,2526,2700,2880,3066,3257,3454,3657,3866,4081,4302,4514,4700,4813,4646,4437,4212,3991,3776,3567,3364,3167,2976,2791,2612,2439,2272,2111,1956,1807,1664,1527,1396,1271,1152,1039,932,831,736,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,742,837,938,1045,1158,1277,1402,1533,1670,1813,1962,2117,2278,2445,2618,2797,2982,3173,3370,3573,3782,3997,4218,4443,4258,4037,3822,3613,3410,3213,3022,2837,2658,2485,2318,2157,2002,1853,1710,1573,1442,1317,1198,1084,976,874,778,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,776,872,974,1082,1195,1314,1439,1570,1707,1850,1999,2154,2315,2482,2655,2834,3019,3210,3407,3610,3819,4034,4255,4475,4672,4826,4963,5082,5183,5117,5007,4879,4736,4549,4336,4114,3898,3688,3484,3286,3094,2908,2728,2554,2386,2224,2068,1918,1774,1635,1387,1138,1255,1269,1276,1283,1290,1297,1304,1311,1318,1324,1329,1334,1339,1344,1349,1354,1359,1364,1368,1371,1373,1501,1634,1772,1916,2066,2222,2384,2552,2726,2906,3092,3284,3482,3686,3896,4112,4334,4547,4734,4877,5005,5115,5209,5290,5355,5401,5416,5412,5408,5404,5403,5402,5387,5333,5260,5169,5060,4933,4790,4609,4396,4174,3958,3747,3355,2962,3151,3165,3172,3179,3186,3193,3200,3207,3214,3221,3228,3235,3242,3249,3256,3262,3267,3272,3277,3282,3287,3292,3297,3302,3307,3312,3317,3322,3327,3332,3336,3339,3341,3541,3746,3956,4172,4394,4607,4659,4447,4215,3987,3757,3347,3546,3742,3737,3732,3727,3722,3717,3712,3707,3702,3697,3692,3687,3682,3677,3672,3667,3660,3653,3646,3639,3632,3625,3618,3611,3604,3597,3590,3583,3576,3569,3562,3547,3346,3545,3752,3964,4181,4403,4616,4796,4939,5066,5175,5266,5298,5302,5306,5310,5314,5318,5322,5326,5330,5334,5337,5291,5211,5110,4994,4860,4711,4518,4299,4071,3849,3633,3423,3219,3021,2829,2643,2463,2289,2121,1959,1803,1645,1379,1506,1630,1625,1620,1615,1610,1605,1600,1595,1590,1585,1578,1571,1564,1557,1550,1543,1536,1529,1522,1507,1378,1505,1640,1780,1925,2075,2231,2393,2561,2735,2915,3101,3293,3491,3695,3905,4121,4343,4556,4743,4886,5014,5124,5216,5184,5083,4964,4827,4674,4479,4261,4040,3825,3616,3413,3216,3025,2840,2661,2488,2321,2160,2005,1856,1713,1576,1445,1320,1197,1077,963,855,753,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,785,881,983,1091,1205,1325,1451,1582,1719,1862,2011,2166,2327,2494,2667,2846,3031,3222,3419,3622,3831,4046,4267,4434,4209,3988,3773,3564,3361,3164,2973,2788,2609,2436,2269,2108,1953,1804,1661,1524,1393,1268,1149,1036,929,828,733,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,730,825,926,1033,1146,1265,1390,1521,1658,1801,1950,2105,2266,2433,2606,2785,2970,3161,3358,3561,3770,3985,4206,4431,4642,4691,4505,4293,4072,3857,3648,3445,3248,3057,2872,2693,2519,2351,2189,2033,1883,1739,1601,1469,1343,1223,1109,1001,899,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,897,999,1107,1221,1341,1467,1599,1737,1881,2031,2187,2349,2517,2690,2869,3054,3245,3442,3645,3854,4069,4290,4502,4688,4836,4973,5092,5193,5276,5341,5360,5305,5232,5141,5032,4904,4761,4574,4361,4139,3923,3713,3509,3311,3119,2933,2753,2579,2410,2102,1793,1940,1954,1961,1968,1975,1982,1989,1996,2003,2010,2017,2024,2029,2034,2039,2044,2049,2054,2059,2064,2069,2074,2079,2083,2086,2088,2246,2409,2577,2751,2931,3117,3309,3507,3711,3921,4137,4359,4572,4759,4902,5030,5071,5067,5063,5059,5055,5051,5047,5043,5039,5035,5031,5026,5021,5016,5011,5006,5000,4993,4986,4982,4978,4952,4809,4428,3977,4196,4210,4217,4224,4231,4238,4245,4252,4259,4266,4273,4280,4287,4294,4301,4308,4315,4322,4327,4332,4337,4342,4347,4352,4357,4362,4367,4372,4377,4382,4387,4392,4397,4402,4407,4411,4414,4416,4631,4676,4477,4250,4022,3800,3584,3374,3170,2972,2772,2422,2591,2757,2752,2747,2742,2737,2732,2727,2722,2717,2712,2707,2702,2697,2691,2684,2677,2670,2663,2656,2649,2642,2635,2628,2621,2614,2607,2592,2421,2590,2767,2949,3136,3328,3526,3730,3940,4156,4378,4591,4776,4919,5046,5155,5246,5319,5374,5411,5422,5393,5346,5281,5198,5097,4980,4850,4708,4522,4310,4089,3874,3665,3458,3254,3056,2864,2678,2498,2324,2156,1994,1838,1688,1544,1406,1274,1148,1020,814,911,1005,1000,995,990,985,980,975,969,962,955,948,941,934,927,912,813,910,1015,1125,1240,1360,1486,1618,1756,1900,2050,2206,2368,2536,2710,2890,3076,3268,3466,3670,3880,4095,4316,4528,4714,4856,4954,4817,4656,4451,4226,4005,3790,3581,3378,3181,2990,2805,2626,2453,2286,2125,1970,1821,1678,1541,1410,1285,1166,1053,946,845,750,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,756,851,952,1059,1172,1291,1416,1547,1684,1827,1976,2131,2292,2459,2632,2811,2996,3187,3384,3587,3796,4011,4232,4457,4244,4023,3808,3599,3396,3199,3008,2823,2644,2471,2304,2143,1988,1839,1696,1559,1428,1303,1184,1071,964,863,768,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,765,860,961,1068,1181,1300,1425,1556,1693,1836,1985,2140,2301,2468,2641,2820,3005,3196,3393,3596,3805,4020,4241,4465,4666,4822,4959,5078,4995,4868,4726,4539,4326,4104,3888,3678,3474,3276,3084,2898,2718,2544,2376,2214,2058,1908,1764,1626,1494,1367,1143,918,1023,1037,1044,1051,1058,1065,1072,1079,1085,1090,1095,1100,1105,1110,1115,1120,1124,1127,1129,1245,1366,1492,1624,1762,1906,2056,2212,2374,2542,2716,2896,3082,3274,3472,3676,3886,4102,4324,4537,4723,4865,4992,5104,5203,5286,5351,5398,5427,5437,5435,5417,5380,5325,5252,5161,5052,4925,4782,4599,4386,4164,3948,3738,3534,3335,2967,2598,2775,2789,2796,2803,2810,2817,2824,2831,2838,2845,2852,2859,2866,2873,2879,2884,2889,2894,2899,2904,2909,2914,2919,2924,2929,2934,2939,2944,2948,2951,2953,3141,3334,3532,3736,3946,4162,4384,4597,4649,4433,4193,3759,3970,4178,4173,4168,4163,4158,4153,4148,4143,4138,4133,4128,4123,4118,4113,4108,4103,4098,4091,4084,4077,4070,4063,4056,4049,4042,4035,4028,4021,4014,4007,4000,3993,3986,3971,3758,3969,4188,4412,4626,4804,4947,5074,5112,5118,5123,5128,5133,5138,5142,5146,5150,5154,5158,5162,5166,5170,5174,5178,5181,5102,4983,4846,4697,4504,4285,4057,3835,3619,3409,3205,3007,2815,2629,2449,2275,2107,1937,1647,1786,1922,1917,1912,1907,1902,1897,1892,1887,1882,1877,1872,1865,1858,1851,1844,1837,1830,1823,1816,1809,1802,1787,1646,1785,1932,2084,2241,2403,2571,2745,2925,3111,3303,3501,3705,3915,4131,4353,4566,4753,4896,5024,5134,5226,5299,5271,5188,5087,4968,4831,4680,4489,4275,4054,3839,3630,3427,3230,3039,2854,2675,2502,2335,2174,2019,1870,1723,1579,1441,1309,1183,1063,949,841,739,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,795,891,993,1101,1215,1335,1461,1593,1731,1875,2025,2180,2341,2508,2681,2860,3045,3236,3433,3636,3845,4060,4281,4197,3976,3761,3552,3349,3152,2961,2776,2597,2424,2257,2096,1941,1792,1649,1512,1381,1256,1137,1024,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)Polycyclic presentation:
2,3,5437|c^a=c^5436,c^b=c^2271912 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32622, 1);