Finite Group DatabasebetaFinite Group Databasebeta| order | 36258 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D6043⋊C3 |
| unambiguous semidirect product | (C6043⋊C3)⋊C2 |
| Fitting, socle, derived, Sylow | C6043 ⋊ C2×C3 |
| Hall | C6043⋊C3 ⋊ C2 |
| Hall | D6043 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C6043 | C2×C3 |
| LCS | G ▷ C6043 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6043 |
| transitive degree | μt(G) = 6043 |
A minimal faithful permutation representation — 2 generators on 6043 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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,1665,1808,1957,2112,2273,2440,2613,2792,2977,3168,3365,3568,3777,3992,4213,4440,4672,4893,4678,4446,4219,3998,3783,3574,3371,3174,2983,2798,2619,2446,2279,2118,1963,1814,1671,1534,1403,1278,1159,1046,939,838,743,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,749,844,945,1052,1165,1284,1409,1540,1677,1820,1969,2124,2285,2452,2625,2804,2989,3180,3377,3580,3789,4004,4225,4452,4684,4903,5093,5124,4938,4725,4495,4268,4047,3832,3623,3420,3223,3032,2847,2668,2495,2328,2167,2012,1863,1720,1583,1448,1316,1190,1070,956,848,746,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,790,886,988,1096,1210,1330,1456,1588,1726,1869,2018,2173,2334,2501,2674,2853,3038,3229,3426,3629,3838,4053,4274,4501,4731,4944,5130,5295,5444,5575,5688,5608,5481,5336,5173,4987,4774,4544,4317,4096,3877,3661,3451,3247,3049,2857,2671,2491,2317,2149,1987,1831,1681,1537,1399,1267,1133,915,1018,1118,1113,1108,1103,1098,1093,1088,1083,1076,1069,1062,1055,1048,1041,1034,1019,914,1017,1128,1244,1365,1491,1623,1761,1905,2055,2211,2373,2541,2715,2895,3081,3273,3471,3675,3885,4101,4323,4550,4780,4993,5179,5342,5487,5614,5723,5818,5895,5954,5976,5912,5832,5734,5618,5484,5332,5162,4969,4749,4512,4278,4050,3828,3612,3402,3198,3000,2808,2622,2442,2268,2092,1790,1935,2077,2072,2067,2062,2057,2052,2047,2042,2037,2032,2027,2021,2014,2007,2000,1993,1986,1979,1972,1965,1958,1951,1936,1789,1934,2087,2245,2408,2576,2750,2930,3116,3308,3506,3710,3920,4136,4358,4586,4817,5031,5218,5382,5528,5656,5766,5858,5932,5953,5958,5963,5968,5973,5978,5982,5987,5945,5878,5795,5695,5577,5441,5287,5115,4920,4700,4463,4229,4001,3779,3563,3345,2959,3146,3330,3325,3320,3315,3310,3305,3300,3295,3290,3285,3280,3275,3270,3265,3260,3253,3246,3239,3232,3225,3218,3211,3204,3197,3190,3183,3176,3169,3162,3147,2958,3145,3340,3540,3745,3955,4171,4393,4621,4852,5066,5092,5097,5102,5107,5112,5117,5122,5129,5136,5143,5150,5157,5164,5171,5178,5185,5192,5197,5202,5207,5212,5217,5222,5227,5232,5237,5242,5247,5252,5084,4882,4422,4651,4872,4867,4863,4859,4854,4849,4844,4839,4834,4829,4824,4819,4814,4809,4804,4799,4794,4789,4783,4776,4769,4762,4755,4748,4741,4734,4727,4720,4713,4706,4699,4694,4687,4680,4673,4666,4652,4421,4650,4864,4634,4406,4183,3767,3350,3551,3565,3572,3579,3586,3593,3600,3607,3614,3621,3628,3635,3642,3649,3656,3663,3669,3674,3679,3684,3689,3694,3699,3704,3709,3714,3719,3724,3729,3734,3739,3744,3748,3751,3753,3965,4182,4404,4632,4862,5073,5255,5415,5557,5681,5787,5865,5860,5855,5850,5845,5838,5831,5824,5819,5814,5809,5804,5799,5794,5790,5788,5779,5669,5541,5395,5231,5044,4830,4599,4371,4149,3933,3723,3519,3321,3129,2943,2762,2430,2097,2256,2270,2277,2284,2291,2298,2305,2312,2319,2326,2333,2340,2347,2352,2357,2362,2367,2372,2377,2382,2387,2392,2397,2402,2407,2411,2414,2416,2586,2761,2941,3127,3319,3517,3721,3931,4147,4369,4597,4828,5042,5229,5393,5539,5667,5777,5868,5938,5990,6024,6032,6008,5967,5908,5833,5742,5633,5506,5360,5196,5009,4795,4564,4336,4114,3898,3688,3484,3286,3094,2908,2728,2554,2386,2224,2068,1918,1774,1635,1387,1138,1255,1269,1276,1283,1290,1297,1304,1311,1318,1324,1329,1334,1339,1344,1349,1354,1359,1364,1368,1371,1373,1501,1634,1772,1916,2066,2222,2384,2552,2726,2906,3092,3284,3482,3686,3896,4112,4334,4562,4793,5007,5194,5358,5503,5630,5739,5830,5796,5701,5588,5457,5312,5149,4963,4750,4520,4293,4072,3857,3648,3445,3248,3057,2872,2693,2519,2351,2189,2033,1883,1739,1601,1469,1343,1223,1109,1001,899,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,897,999,1107,1221,1341,1467,1599,1737,1881,2031,2187,2349,2517,2690,2869,3054,3245,3442,3645,3854,4069,4290,4517,4747,4960,5146,5309,5423,5273,5106,4916,4701,4471,4244,4023,3808,3599,3396,3199,3008,2823,2644,2471,2304,2143,1988,1839,1696,1559,1428,1303,1184,1071,964,863,768,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,765,860,961,1068,1181,1300,1425,1556,1693,1836,1985,2140,2301,2468,2641,2820,3005,3196,3393,3596,3805,4020,4241,4468,4657,4424,4197,3976,3761,3552,3349,3152,2961,2776,2597,2424,2257,2096,1941,1792,1649,1512,1381,1256,1137,1024,917,816,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)Polycyclic presentation:
2,3,6043|c^a=c^6042,c^b=c^17151013 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36258, 1);