Dic₄₅₄₂ #72216009

order	18168
automorphism	Aut(G) order 27470016
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₅₄₂

Splittings

Hall	C₃×C₇₅₇ ⋊ Q₈
Sylow	C₇₅₇ ⋊ Dic₆
Sylow	C₃ ⋊ Dic₁₅₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₂₇₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₇₅₇	C₂²
Fitting	C₃×C₄×C₇₅₇	C₂
hypercenter	C₄	D₂₂₇₁
hypocenter	C₃×C₇₅₇	Q₈
LCS	G ▷ C₂×C₃×C₇₅₇ ▷ C₃×C₇₅₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 768
transitive degree	μ_t(G) = 18168

A minimal faithful permutation representation — 2 generators on 768 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);(1,5,3,7)(2,6,4,8)(10,11)(13,768)(14,767)(15,766)(16,765)(17,764)(18,763)(19,762)(20,761)(21,760)(22,759)(23,758)(24,757)(25,756)(26,755)(27,754)(28,753)(29,752)(30,751)(31,750)(32,749)(33,748)(34,747)(35,746)(36,745)(37,744)(38,743)(39,742)(40,741)(41,740)(42,739)(43,738)(44,737)(45,736)(46,735)(47,734)(48,733)(49,732)(50,731)(51,730)(52,729)(53,728)(54,727)(55,726)(56,725)(57,724)(58,723)(59,722)(60,721)(61,720)(62,719)(63,718)(64,717)(65,716)(66,715)(67,714)(68,713)(69,712)(70,711)(71,710)(72,709)(73,708)(74,707)(75,706)(76,705)(77,704)(78,703)(79,702)(80,701)(81,700)(82,699)(83,698)(84,697)(85,696)(86,695)(87,694)(88,693)(89,692)(90,691)(91,690)(92,689)(93,688)(94,687)(95,686)(96,685)(97,684)(98,683)(99,682)(100,681)(101,680)(102,679)(103,678)(104,677)(105,676)(106,675)(107,674)(108,673)(109,672)(110,671)(111,670)(112,669)(113,668)(114,667)(115,666)(116,665)(117,664)(118,663)(119,662)(120,661)(121,660)(122,659)(123,658)(124,657)(125,656)(126,655)(127,654)(128,653)(129,652)(130,651)(131,650)(132,649)(133,648)(134,647)(135,646)(136,645)(137,644)(138,643)(139,642)(140,641)(141,640)(142,639)(143,638)(144,637)(145,636)(146,635)(147,634)(148,633)(149,632)(150,631)(151,630)(152,629)(153,628)(154,627)(155,626)(156,625)(157,624)(158,623)(159,622)(160,621)(161,620)(162,619)(163,618)(164,617)(165,616)(166,615)(167,614)(168,613)(169,612)(170,611)(171,610)(172,609)(173,608)(174,607)(175,606)(176,605)(177,604)(178,603)(179,602)(180,601)(181,600)(182,599)(183,598)(184,597)(185,596)(186,595)(187,594)(188,593)(189,592)(190,591)(191,590)(192,589)(193,588)(194,587)(195,586)(196,585)(197,584)(198,583)(199,582)(200,581)(201,580)(202,579)(203,578)(204,577)(205,576)(206,575)(207,574)(208,573)(209,572)(210,571)(211,570)(212,569)(213,568)(214,567)(215,566)(216,565)(217,564)(218,563)(219,562)(220,561)(221,560)(222,559)(223,558)(224,557)(225,556)(226,555)(227,554)(228,553)(229,552)(230,551)(231,550)(232,549)(233,548)(234,547)(235,546)(236,545)(237,544)(238,543)(239,542)(240,541)(241,540)(242,539)(243,538)(244,537)(245,536)(246,535)(247,534)(248,533)(249,532)(250,531)(251,530)(252,529)(253,528)(254,527)(255,526)(256,525)(257,524)(258,523)(259,522)(260,521)(261,520)(262,519)(263,518)(264,517)(265,516)(266,515)(267,514)(268,513)(269,512)(270,511)(271,510)(272,509)(273,508)(274,507)(275,506)(276,505)(277,504)(278,503)(279,502)(280,501)(281,500)(282,499)(283,498)(284,497)(285,496)(286,495)(287,494)(288,493)(289,492)(290,491)(291,490)(292,489)(293,488)(294,487)(295,486)(296,485)(297,484)(298,483)(299,482)(300,481)(301,480)(302,479)(303,478)(304,477)(305,476)(306,475)(307,474)(308,473)(309,472)(310,471)(311,470)(312,469)(313,468)(314,467)(315,466)(316,465)(317,464)(318,463)(319,462)(320,461)(321,460)(322,459)(323,458)(324,457)(325,456)(326,455)(327,454)(328,453)(329,452)(330,451)(331,450)(332,449)(333,448)(334,447)(335,446)(336,445)(337,444)(338,443)(339,442)(340,441)(341,440)(342,439)(343,438)(344,437)(345,436)(346,435)(347,434)(348,433)(349,432)(350,431)(351,430)(352,429)(353,428)(354,427)(355,426)(356,425)(357,424)(358,423)(359,422)(360,421)(361,420)(362,419)(363,418)(364,417)(365,416)(366,415)(367,414)(368,413)(369,412)(370,411)(371,410)(372,409)(373,408)(374,407)(375,406)(376,405)(377,404)(378,403)(379,402)(380,401)(381,400)(382,399)(383,398)(384,397)(385,396)(386,395)(387,394)(388,393)(389,392)(390,391)

Polycyclic presentation:

2^3,3,757|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^756

Conjugacy classes & character table

4545 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₈₄, Dic₁₃₆₂₆
maximal normal subgroup of	Dic₉₀₈₄
maximal quotient of	Dic₁₃₆₂₆

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, 3, 757]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^756);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4542 #72216009