Dic₁₄₁₈₀ #72290672

order	56720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₈₀

Splittings

Hall	C₅×C₇₀₉ ⋊ Q₁₆
Sylow	C₇₀₉ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₂₈₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₄₁₈₀
commutator, cosocle	C₄×C₅×C₇₀₉	C₂²
Frattini	C₄	C₂×D₃₅₄₅
Fitting	C₅×C₈×C₇₀₉	C₂
socle, prime radical	C₂×C₅×C₇₀₉	D₄
hypercenter	C₈	D₃₅₄₅
hypocenter	C₅×C₇₀₉	Q₁₆
LCS	G ▷ C₄×C₅×C₇₀₉ ▷ C₂×C₅×C₇₀₉ ▷ C₅×C₇₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 730
transitive degree	μ_t(G) = 56720

A minimal faithful permutation representation — 2 generators on 730 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,730)(24,729)(25,728)(26,727)(27,726)(28,725)(29,724)(30,723)(31,722)(32,721)(33,720)(34,719)(35,718)(36,717)(37,716)(38,715)(39,714)(40,713)(41,712)(42,711)(43,710)(44,709)(45,708)(46,707)(47,706)(48,705)(49,704)(50,703)(51,702)(52,701)(53,700)(54,699)(55,698)(56,697)(57,696)(58,695)(59,694)(60,693)(61,692)(62,691)(63,690)(64,689)(65,688)(66,687)(67,686)(68,685)(69,684)(70,683)(71,682)(72,681)(73,680)(74,679)(75,678)(76,677)(77,676)(78,675)(79,674)(80,673)(81,672)(82,671)(83,670)(84,669)(85,668)(86,667)(87,666)(88,665)(89,664)(90,663)(91,662)(92,661)(93,660)(94,659)(95,658)(96,657)(97,656)(98,655)(99,654)(100,653)(101,652)(102,651)(103,650)(104,649)(105,648)(106,647)(107,646)(108,645)(109,644)(110,643)(111,642)(112,641)(113,640)(114,639)(115,638)(116,637)(117,636)(118,635)(119,634)(120,633)(121,632)(122,631)(123,630)(124,629)(125,628)(126,627)(127,626)(128,625)(129,624)(130,623)(131,622)(132,621)(133,620)(134,619)(135,618)(136,617)(137,616)(138,615)(139,614)(140,613)(141,612)(142,611)(143,610)(144,609)(145,608)(146,607)(147,606)(148,605)(149,604)(150,603)(151,602)(152,601)(153,600)(154,599)(155,598)(156,597)(157,596)(158,595)(159,594)(160,593)(161,592)(162,591)(163,590)(164,589)(165,588)(166,587)(167,586)(168,585)(169,584)(170,583)(171,582)(172,581)(173,580)(174,579)(175,578)(176,577)(177,576)(178,575)(179,574)(180,573)(181,572)(182,571)(183,570)(184,569)(185,568)(186,567)(187,566)(188,565)(189,564)(190,563)(191,562)(192,561)(193,560)(194,559)(195,558)(196,557)(197,556)(198,555)(199,554)(200,553)(201,552)(202,551)(203,550)(204,549)(205,548)(206,547)(207,546)(208,545)(209,544)(210,543)(211,542)(212,541)(213,540)(214,539)(215,538)(216,537)(217,536)(218,535)(219,534)(220,533)(221,532)(222,531)(223,530)(224,529)(225,528)(226,527)(227,526)(228,525)(229,524)(230,523)(231,522)(232,521)(233,520)(234,519)(235,518)(236,517)(237,516)(238,515)(239,514)(240,513)(241,512)(242,511)(243,510)(244,509)(245,508)(246,507)(247,506)(248,505)(249,504)(250,503)(251,502)(252,501)(253,500)(254,499)(255,498)(256,497)(257,496)(258,495)(259,494)(260,493)(261,492)(262,491)(263,490)(264,489)(265,488)(266,487)(267,486)(268,485)(269,484)(270,483)(271,482)(272,481)(273,480)(274,479)(275,478)(276,477)(277,476)(278,475)(279,474)(280,473)(281,472)(282,471)(283,470)(284,469)(285,468)(286,467)(287,466)(288,465)(289,464)(290,463)(291,462)(292,461)(293,460)(294,459)(295,458)(296,457)(297,456)(298,455)(299,454)(300,453)(301,452)(302,451)(303,450)(304,449)(305,448)(306,447)(307,446)(308,445)(309,444)(310,443)(311,442)(312,441)(313,440)(314,439)(315,438)(316,437)(317,436)(318,435)(319,434)(320,433)(321,432)(322,431)(323,430)(324,429)(325,428)(326,427)(327,426)(328,425)(329,424)(330,423)(331,422)(332,421)(333,420)(334,419)(335,418)(336,417)(337,416)(338,415)(339,414)(340,413)(341,412)(342,411)(343,410)(344,409)(345,408)(346,407)(347,406)(348,405)(349,404)(350,403)(351,402)(352,401)(353,400)(354,399)(355,398)(356,397)(357,396)(358,395)(359,394)(360,393)(361,392)(362,391)(363,390)(364,389)(365,388)(366,387)(367,386)(368,385)(369,384)(370,383)(371,382)(372,381)(373,380)(374,379)(375,378)(376,377)

Polycyclic presentation:

2^4,5,709|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^708

Conjugacy classes & character table

14183 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", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 709]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^708);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14180 #72290672