C₃×D₅₁₅₅ #44525800

order	30930
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅₁₅₅
unambiguous semidirect product	C₅⋊(C₃×D₁₀₃₁)
unambiguous semidirect product	C₁₀₃₁⋊(C₃×D₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₀₃₁ ⋊ C₂
derived, Hall	C₅×C₁₀₃₁ ⋊ C₂×C₃
Hall	C₃×C₁₀₃₁ ⋊ D₅
Sylow	C₁₀₃₁ ⋊ C₃×D₅
Hall	C₃×C₅ ⋊ D₁₀₃₁
Sylow	C₅ ⋊ C₃×D₁₀₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₅₁₅₅
commutator, cosocle, hypocenter	C₅×C₁₀₃₁	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₀₃₁	C₂
LCS	G ▷ C₅×C₁₀₃₁
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1039
transitive degree	μ_t(G) = 15465

A minimal faithful permutation representation — 2 generators on 1039 points:

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

Conjugacy classes & character table

7737 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30930, 5);

C3×D5155 #44525800