Dic₅₃₂₇ #36319211

order	21308
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₃₂₇
unambiguous semidirect product	C₇⋊Dic₇₆₁

Splittings

derived, Hall	C₇×C₇₆₁ ⋊ C₄
Sylow	C₇₆₁ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₇₆₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₃₂₇
commutator, hypocenter	C₇×C₇₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₇₆₁	C₂
LCS	G ▷ C₇×C₇₆₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 772
transitive degree	μ_t(G) = 21308

A minimal faithful permutation representation — 2 generators on 772 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,772)(14,771)(15,770)(16,769)(17,768)(18,767)(19,766)(20,765)(21,764)(22,763)(23,762)(24,761)(25,760)(26,759)(27,758)(28,757)(29,756)(30,755)(31,754)(32,753)(33,752)(34,751)(35,750)(36,749)(37,748)(38,747)(39,746)(40,745)(41,744)(42,743)(43,742)(44,741)(45,740)(46,739)(47,738)(48,737)(49,736)(50,735)(51,734)(52,733)(53,732)(54,731)(55,730)(56,729)(57,728)(58,727)(59,726)(60,725)(61,724)(62,723)(63,722)(64,721)(65,720)(66,719)(67,718)(68,717)(69,716)(70,715)(71,714)(72,713)(73,712)(74,711)(75,710)(76,709)(77,708)(78,707)(79,706)(80,705)(81,704)(82,703)(83,702)(84,701)(85,700)(86,699)(87,698)(88,697)(89,696)(90,695)(91,694)(92,693)(93,692)(94,691)(95,690)(96,689)(97,688)(98,687)(99,686)(100,685)(101,684)(102,683)(103,682)(104,681)(105,680)(106,679)(107,678)(108,677)(109,676)(110,675)(111,674)(112,673)(113,672)(114,671)(115,670)(116,669)(117,668)(118,667)(119,666)(120,665)(121,664)(122,663)(123,662)(124,661)(125,660)(126,659)(127,658)(128,657)(129,656)(130,655)(131,654)(132,653)(133,652)(134,651)(135,650)(136,649)(137,648)(138,647)(139,646)(140,645)(141,644)(142,643)(143,642)(144,641)(145,640)(146,639)(147,638)(148,637)(149,636)(150,635)(151,634)(152,633)(153,632)(154,631)(155,630)(156,629)(157,628)(158,627)(159,626)(160,625)(161,624)(162,623)(163,622)(164,621)(165,620)(166,619)(167,618)(168,617)(169,616)(170,615)(171,614)(172,613)(173,612)(174,611)(175,610)(176,609)(177,608)(178,607)(179,606)(180,605)(181,604)(182,603)(183,602)(184,601)(185,600)(186,599)(187,598)(188,597)(189,596)(190,595)(191,594)(192,593)(193,592)(194,591)(195,590)(196,589)(197,588)(198,587)(199,586)(200,585)(201,584)(202,583)(203,582)(204,581)(205,580)(206,579)(207,578)(208,577)(209,576)(210,575)(211,574)(212,573)(213,572)(214,571)(215,570)(216,569)(217,568)(218,567)(219,566)(220,565)(221,564)(222,563)(223,562)(224,561)(225,560)(226,559)(227,558)(228,557)(229,556)(230,555)(231,554)(232,553)(233,552)(234,551)(235,550)(236,549)(237,548)(238,547)(239,546)(240,545)(241,544)(242,543)(243,542)(244,541)(245,540)(246,539)(247,538)(248,537)(249,536)(250,535)(251,534)(252,533)(253,532)(254,531)(255,530)(256,529)(257,528)(258,527)(259,526)(260,525)(261,524)(262,523)(263,522)(264,521)(265,520)(266,519)(267,518)(268,517)(269,516)(270,515)(271,514)(272,513)(273,512)(274,511)(275,510)(276,509)(277,508)(278,507)(279,506)(280,505)(281,504)(282,503)(283,502)(284,501)(285,500)(286,499)(287,498)(288,497)(289,496)(290,495)(291,494)(292,493)(293,492)(294,491)(295,490)(296,489)(297,488)(298,487)(299,486)(300,485)(301,484)(302,483)(303,482)(304,481)(305,480)(306,479)(307,478)(308,477)(309,476)(310,475)(311,474)(312,473)(313,472)(314,471)(315,470)(316,469)(317,468)(318,467)(319,466)(320,465)(321,464)(322,463)(323,462)(324,461)(325,460)(326,459)(327,458)(328,457)(329,456)(330,455)(331,454)(332,453)(333,452)(334,451)(335,450)(336,449)(337,448)(338,447)(339,446)(340,445)(341,444)(342,443)(343,442)(344,441)(345,440)(346,439)(347,438)(348,437)(349,436)(350,435)(351,434)(352,433)(353,432)(354,431)(355,430)(356,429)(357,428)(358,427)(359,426)(360,425)(361,424)(362,423)(363,422)(364,421)(365,420)(366,419)(367,418)(368,417)(369,416)(370,415)(371,414)(372,413)(373,412)(374,411)(375,410)(376,409)(377,408)(378,407)(379,406)(380,405)(381,404)(382,403)(383,402)(384,401)(385,400)(386,399)(387,398)(388,397)(389,396)(390,395)(391,394)(392,393)

Polycyclic presentation:

2,7,761,2|a^2=d,b^a=b^6,c^a=c^760

Conjugacy classes & character table

5330 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₆₅₄, Dic₁₅₉₈₁
maximal normal subgroup of	Dic₁₀₆₅₄
maximal quotient of	Dic₁₅₉₈₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21308, 4);

Dic5327 #36319211