Dic₅₁₅₅ #35207348

order	20620
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₅₅
unambiguous semidirect product	C₁₀₃₁⋊Dic₅

Splittings

derived, Hall	C₅×C₁₀₃₁ ⋊ C₄
Sylow	C₁₀₃₁ ⋊ Dic₅
Sylow	C₅ ⋊ Dic₁₀₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₅₅
commutator, hypocenter	C₅×C₁₀₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₀₃₁	C₂
LCS	G ▷ C₅×C₁₀₃₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1040
transitive degree	μ_t(G) = 20620

A minimal faithful permutation representation — 2 generators on 1040 points:

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

Polycyclic presentation:

2,5,1031,2|a^2=d,b^a=b^4,c^a=c^1030

Conjugacy classes & character table

5158 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₁₀, Dic₁₅₄₆₅
maximal normal subgroup of	Dic₁₀₃₁₀
maximal quotient of	Dic₁₅₄₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20620, 6);

Dic5155 #35207348