D₁₅₃₆₀ #72272082

order	30720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₃₆₀

Splittings

Fitting	C₃×C₅×C₁₀₂₄ ⋊ C₂
Hall	C₃×C₅ ⋊ D₁₀₂₄
Sylow	C₅ ⋊ D₃₀₇₂
Sylow	C₃ ⋊ D₅₁₂₀

Subgroups

	subgroup	G/H
center	C₂	D₇₆₈₀
commutator, cosocle	C₃×C₅×C₅₁₂	C₂²
Frattini	C₅₁₂	C₂×D₁₅
Fitting	C₃×C₅×C₁₀₂₄	C₂
socle	C₂×C₃×C₅	D₅₁₂
hypercenter	C₁₀₂₄	D₁₅
hypocenter	C₃×C₅	D₁₀₂₄
LCS	G ▷ C₃×C₅×C₅₁₂ ▷ C₃×C₅×C₂₅₆ ▷ C₃×C₅×C₁₂₈ ▷ C₃×C₅×C₆₄ ▷ C₃×C₅×C₃₂ ▷ C₃×C₅×C₁₆ ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈ ◁ C₂₅₆ ◁ C₅₁₂ ◁ C₁₀₂₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1032
transitive degree	μ_t(G) = 15360

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

Polycyclic presentation:

2^11,3,5|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=jk,j^2=k,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij,j^a=jk,l^a=l^2,m^a=m^4

Conjugacy classes & character table

7683 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₇₂₀
maximal normal subgroup of	D₃₀₇₂₀
maximal quotient of	Dic₁₅₃₆₀, D₃₀₇₂₀

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 12, 1, F.12^2);;
gap> SetConjugate(coll, 13, 1, F.13^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15360 #72272082