Dic₁₁₅₇₈ #72277181

order	46312
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₅₇₈

Splittings

Hall	C₇×C₈₂₇ ⋊ Q₈
Sylow	C₈₂₇ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₁₆₅₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₇₈₉
commutator, socle, prime radical, cosocle	C₂×C₇×C₈₂₇	C₂²
Fitting	C₄×C₇×C₈₂₇	C₂
hypercenter	C₄	D₅₇₈₉
hypocenter	C₇×C₈₂₇	Q₈
LCS	G ▷ C₂×C₇×C₈₂₇ ▷ C₇×C₈₂₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 842
transitive degree	μ_t(G) = 46312

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

Polycyclic presentation:

2^3,7,827|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^826

Conjugacy classes & character table

11581 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, 827]);;
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.5^826);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11578 #72277181