Finite Group DatabasebetaFinite Group Databasebeta| order | 33870 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(D1129⋊C3) |
| unambiguous semidirect product | (C1129⋊C3)⋊D5 |
| unambiguous semidirect product | D5645⋊C3 |
| Fitting, socle, derived, Hall | C5×C1129 ⋊ C2×C3 |
| Hall | C5×C1129⋊C3 ⋊ C2 |
| Hall | D5645 ⋊ C3 |
| Hall | C1129⋊C3 ⋊ D5 |
| Sylow | C1129 ⋊ C3×D5 |
| Sylow | C5 ⋊ D1129⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C1129 | C2×C3 |
| LCS | G ▷ C5×C1129 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1134 |
| transitive degree | μt(G) = 5645 |
A minimal faithful permutation representation — 2 generators on 1134 points:
(2,3)(4,5)(7,11,12,8,9,13)(10,19,20,14,15,29)(16,32,33,22,23,46)(17,31,27,25,21,18)(24,51,52,35,36,69)(26,49,50,38,34,43)(28,30,48,40,41,44)(37,74,75,54,55,98)(39,72,73,57,53,66)(42,47,71,59,60,67)(45,87,85,63,64,62)(56,103,104,77,78,133)(58,101,102,80,76,95)(61,70,100,82,83,96)(65,116,117,88,89,91)(68,122,114,92,93,86)(79,138,139,106,107,174)(81,136,137,109,105,130)(84,99,135,111,112,131)(90,151,152,118,119,126)(94,158,156,123,124,121)(97,163,149,127,128,115)(108,179,180,141,142,221)(110,177,178,144,140,171)(113,134,176,146,147,172)(120,192,193,153,154,167)(125,199,200,159,160,162)(129,205,197,164,165,157)(132,210,190,168,169,150)(143,226,227,182,183,274)(145,224,225,185,181,218)(148,175,223,187,188,219)(155,239,240,194,195,214)(161,246,247,201,202,209)(166,253,251,206,207,204)(170,258,244,211,212,198)(173,263,237,215,216,191)(184,279,280,229,230,333)(186,277,278,232,228,271)(189,222,276,234,235,272)(196,292,293,241,242,267)(203,299,300,248,249,262)(208,306,307,254,255,257)(213,312,304,259,260,252)(217,317,297,264,265,245)(220,322,290,268,269,238)(231,338,339,282,283,398)(233,336,337,285,281,330)(236,275,335,287,288,331)(243,351,352,294,295,326)(250,358,359,301,302,321)(256,365,366,308,309,316)(261,372,370,313,314,311)(266,377,363,318,319,305)(270,382,356,323,324,298)(273,387,349,327,328,291)(284,403,404,341,342,469)(286,401,402,344,340,395)(289,334,400,346,347,396)(296,416,417,353,354,391)(303,423,424,360,361,386)(310,430,431,367,368,381)(315,437,438,373,374,376)(320,443,435,378,379,371)(325,448,428,383,384,364)(329,453,421,388,389,357)(332,458,414,392,393,350)(343,474,475,406,407,546)(345,472,473,409,405,466)(348,399,471,411,412,467)(355,487,488,418,419,462)(362,494,495,425,426,457)(369,501,502,432,433,452)(375,508,509,439,440,447)(380,515,513,444,445,442)(385,520,506,449,450,436)(390,525,499,454,455,429)(394,530,492,459,460,422)(397,535,485,463,464,415)(408,551,552,477,478,629)(410,549,550,480,476,543)(413,470,548,482,483,544)(420,564,565,489,490,539)(427,571,572,496,497,534)(434,578,579,503,504,529)(441,585,586,510,511,524)(446,592,593,516,517,519)(451,598,590,521,522,514)(456,603,583,526,527,507)(461,608,576,531,532,500)(465,613,569,536,537,493)(468,618,562,540,541,486)(479,634,635,554,555,718)(481,632,633,557,553,626)(484,547,631,559,560,627)(491,647,648,566,567,622)(498,654,655,573,574,617)(505,661,662,580,581,612)(512,668,669,587,588,607)(518,675,676,594,595,602)(523,682,680,599,600,597)(528,687,673,604,605,591)(533,692,666,609,610,584)(538,697,659,614,615,577)(542,702,652,619,620,570)(545,707,645,623,624,563)(556,723,724,637,638,813)(558,721,722,640,636,715)(561,630,720,642,643,716)(568,736,737,649,650,711)(575,743,744,656,657,706)(582,750,751,663,664,701)(589,757,758,670,671,696)(596,764,765,677,678,691)(601,771,772,683,684,686)(606,777,769,688,689,681)(611,782,762,693,694,674)(616,787,755,698,699,667)(621,792,748,703,704,660)(625,797,741,708,709,653)(628,802,734,712,713,646)(639,818,819,726,727,908)(641,816,817,729,725,810)(644,719,815,731,732,811)(651,829,830,738,739,806)(658,834,835,745,746,801)(665,839,840,752,753,796)(672,846,847,759,760,791)(679,853,854,766,767,786)(685,860,861,773,774,781)(690,867,865,778,779,776)(695,872,858,783,784,770)(700,877,851,788,789,763)(705,882,844,793,794,756)(710,887,837,798,799,749)(714,892,832,803,804,742)(717,897,827,807,808,735)(728,836,896,747,821,920)(730,911,912,822,820,905)(733,814,910,824,825,906)(740,913,917,823,831,901)(754,923,924,841,842,891)(761,927,928,848,849,886)(768,934,935,855,856,881)(775,941,942,862,863,876)(780,948,949,868,869,871)(785,954,946,873,874,866)(790,959,939,878,879,859)(795,964,932,883,884,852)(800,969,925,888,889,845)(805,974,921,893,894,838)(809,979,918,898,899,833)(812,983,915,902,903,828)(826,909,978,843,914,990)(850,997,998,929,930,973)(857,1001,1002,936,937,968)(864,1005,1006,943,944,963)(870,1012,1013,950,951,958)(875,1019,1017,955,956,953)(880,1024,1010,960,961,947)(885,1029,1003,965,966,940)(890,1034,999,970,971,933)(895,1039,995,975,976,926)(900,1044,993,980,981,922)(904,1048,991,984,985,919)(907,1043,931,987,988,916)(938,1052,1053,992,989,1038)(945,1060,1061,1007,1008,1033)(952,1064,1065,1014,1015,1028)(957,1068,1069,1020,1021,1023)(962,1074,1066,1025,1026,1018)(967,1079,1062,1030,1031,1011)(972,1084,1058,1035,1036,1004)(977,1089,1056,1040,1041,1000)(982,1093,1054,1045,1046,996)(986,1088,1009,1049,1050,994)(1016,1097,1098,1055,1051,1083)(1022,1105,1106,1070,1071,1078)(1027,1109,1107,1075,1076,1073)(1032,1111,1103,1080,1081,1067)(1037,1116,1101,1085,1086,1063)(1042,1120,1099,1090,1091,1059)(1047,1115,1072,1094,1095,1057)(1077,1124,1125,1100,1096,1110)(1082,1131,1128,1112,1113,1108)(1087,1132,1126,1117,1118,1104)(1092,1130,1114,1121,1122,1102)(1119,1133,1134,1127,1123,1129);(1,2,4,5,3)(6,7,10,16,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,745,656,573,496,425,360,301,248,201,159,123,92,60,23,34,44,46,66,91,121,157,198,245,298,357,422,493,570,653,742,833,919,992,936,855,766,677,594,516,444,378,318,264,215,147,78,105,119,124,128,131,133,171,214,262,316,376,442,514,591,674,763,852,933,1000,1057,1070,1020,955,873,783,693,609,531,459,392,288,183,228,242,249,255,260,265,269,272,274,330,391,457,529,607,691,781,871,953,1018,1067,1104,1127,1117,1085,1035,970,888,798,708,623,483,342,405,419,426,433,440,445,450,455,460,464,467,469,543,622,706,796,886,968,1033,1083,1115,1120,1116,1111,1109,1105,1097,1049,984,902,732,555,636,650,657,664,671,678,684,689,694,699,704,709,713,716,718,810,901,819,910,983,979,974,969,964,959,954,948,941,934,927,923,821,913,911,818,909,930,937,944,951,956,961,966,971,976,981,985,988,990,920,830,722,552,631,707,702,697,692,687,682,675,668,661,654,647,632,551,630,717,809,900,982,1047,1096,1113,1118,1122,1125,1106,1065,1006,935,847,751,655,565,473,339,400,458,453,448,443,437,430,423,416,401,338,399,468,542,621,705,795,885,967,1032,1082,1114,1101,1062,1010,946,865,772,676,586,502,424,352,278,180,223,263,258,253,246,239,224,179,222,273,329,390,456,528,606,690,780,870,952,1016,1054,995,925,844,755,666,583,506,435,370,307,247,193,137,75,100,122,116,101,74,99,132,170,213,261,315,375,441,512,589,672,761,850,931,915,827,734,645,562,485,414,349,290,237,190,149,114,85,50,20,31,19,30,45,65,90,120,155,196,243,296,355,420,491,568,651,740,822,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,9,13,18,28,42,61,84,113,148,189,236,289,348,413,484,561,644,733,826,841,752,663,580,503,432,367,308,254,206,164,127,83,36,53,64,67,69,95,126,162,204,252,305,364,429,500,577,660,749,838,922,994,1007,943,862,773,683,599,521,449,383,323,268,188,107,140,154,160,165,169,172,174,218,267,321,381,447,519,597,681,770,859,940,1004,1059,1100,1075,1025,960,878,788,698,614,536,463,347,230,281,295,302,309,314,319,324,328,331,333,395,462,534,612,696,786,876,958,1023,1073,1108,1129,1133,1121,1090,1040,975,893,803,712,560,407,476,490,497,504,511,517,522,527,532,537,541,544,546,626,711,801,891,973,1038,1088,1093,1089,1084,1079,1074,1068,1064,1060,1052,987,825,638,725,739,746,753,760,767,774,779,784,789,794,799,804,808,811,813,905,912,724,815,897,892,887,882,877,872,867,860,853,846,839,834,829,816,723,814,907,989,1008,1015,1021,1026,1031,1036,1041,1046,1050,1053,998,924,835,737,633,475,548,618,613,608,603,598,592,585,578,571,564,549,474,547,628,714,805,895,977,1042,1092,1123,1134,1128,1107,1069,1013,942,854,758,662,572,488,402,280,335,387,382,377,372,365,358,351,336,279,334,397,465,538,616,700,790,880,962,1027,1077,1099,1058,1003,939,858,769,680,593,509,431,359,293,225,139,176,210,205,199,192,177,138,175,220,270,325,385,451,523,601,685,775,864,945,1009,993,921,837,748,659,576,499,428,363,304,251,200,152,102,52,71,87,72,51,70,97,129,166,208,256,310,369,434,505,582,665,754,843,824,731,642,559,482,411,346,287,234,187,146,111,82,59,40,27,12,11,17,26,39,58,81,110,145,186,233,286,345,410,481,558,641,730,823,738,649,566,489,418,353,294,241,194,153,118,88,63,41,15,21,29,43,62,86,115,150,191,238,291,350,415,486,563,646,735,828,916,929,848,759,670,587,510,439,373,313,259,211,168,112,55,76,89,93,96,98,130,167,209,257,311,371,436,507,584,667,756,845,926,996,1055,1014,950,868,778,688,604,526,454,388,327,235,142,181,195,202,207,212,216,219,221,271,326,386,452,524,602,686,776,866,947,1011,1063,1102,1112,1080,1030,965,883,793,703,619,540,412,283,340,354,361,368,374,379,384,389,393,396,398,466,539,617,701,791,881,963,1028,1078,1110,1130,1132,1131,1124,1094,1045,980,898,807,643,478,553,567,574,581,588,595,600,605,610,615,620,624,627,629,715,806,896,978,1043,1048,1044,1039,1034,1029,1024,1019,1012,1005,1001,997,914,727,820,831,836,842,849,856,863,869,874,879,884,889,894,899,903,906,908,917,817,635,720,802,797,792,787,782,777,771,764,757,750,743,736,721,634,719,812,904,986,1051,1071,1076,1081,1086,1091,1095,1098,1061,1002,928,840,744,648,550,404,471,535,530,525,520,515,508,501,494,487,472,403,470,545,625,710,800,890,972,1037,1087,1119,1126,1103,1066,1017,949,861,765,669,579,495,417,337,227,276,322,317,312,306,299,292,277,226,275,332,394,461,533,611,695,785,875,957,1022,1072,1056,999,932,851,762,673,590,513,438,366,300,240,178,104,135,163,158,151,136,103,134,173,217,266,320,380,446,518,596,679,768,857,938,991,918,832,741,652,569,492,421,356,297,244,197,156,117,73,33,48,49,32,47,68,94,125,161,203,250,303,362,427,498,575,658,747,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,14,8)Polycyclic presentation:
2,3,5,1129|c^a=c^4,d^a=d^1128,d^b=d^387952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33870, 12);