Dic₁₃₆₇₈ #72289865

order	54712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₇₈

Splittings

Hall	C₇×C₉₇₇ ⋊ Q₈
Sylow	C₉₇₇ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₁₉₅₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₃₉
commutator, socle, prime radical, cosocle	C₂×C₇×C₉₇₇	C₂²
Fitting	C₄×C₇×C₉₇₇	C₂
hypercenter	C₄	D₆₈₃₉
hypocenter	C₇×C₉₇₇	Q₈
LCS	G ▷ C₂×C₇×C₉₇₇ ▷ C₇×C₉₇₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 992
transitive degree	μ_t(G) = 54712

A minimal faithful permutation representation — 2 generators on 992 points:

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

Polycyclic presentation:

2^3,7,977|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^976

Conjugacy classes & character table

13681 conjugacy classes. The character table is too large to display.

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 977]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^976);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13678 #72289865