Dic₈₁₂₉ #45654170

order	32516
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₁₂₉
unambiguous semidirect product	C₁₁⋊Dic₇₃₉
unambiguous semidirect product	C₇₃₉⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₇₃₉ ⋊ C₄
Sylow	C₇₃₉ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₇₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₁₂₉
commutator, hypocenter	C₁₁×C₇₃₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₇₃₉	C₂
LCS	G ▷ C₁₁×C₇₃₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 754
transitive degree	μ_t(G) = 32516

A minimal faithful permutation representation — 2 generators on 754 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,754)(18,753)(19,752)(20,751)(21,750)(22,749)(23,748)(24,747)(25,746)(26,745)(27,744)(28,743)(29,742)(30,741)(31,740)(32,739)(33,738)(34,737)(35,736)(36,735)(37,734)(38,733)(39,732)(40,731)(41,730)(42,729)(43,728)(44,727)(45,726)(46,725)(47,724)(48,723)(49,722)(50,721)(51,720)(52,719)(53,718)(54,717)(55,716)(56,715)(57,714)(58,713)(59,712)(60,711)(61,710)(62,709)(63,708)(64,707)(65,706)(66,705)(67,704)(68,703)(69,702)(70,701)(71,700)(72,699)(73,698)(74,697)(75,696)(76,695)(77,694)(78,693)(79,692)(80,691)(81,690)(82,689)(83,688)(84,687)(85,686)(86,685)(87,684)(88,683)(89,682)(90,681)(91,680)(92,679)(93,678)(94,677)(95,676)(96,675)(97,674)(98,673)(99,672)(100,671)(101,670)(102,669)(103,668)(104,667)(105,666)(106,665)(107,664)(108,663)(109,662)(110,661)(111,660)(112,659)(113,658)(114,657)(115,656)(116,655)(117,654)(118,653)(119,652)(120,651)(121,650)(122,649)(123,648)(124,647)(125,646)(126,645)(127,644)(128,643)(129,642)(130,641)(131,640)(132,639)(133,638)(134,637)(135,636)(136,635)(137,634)(138,633)(139,632)(140,631)(141,630)(142,629)(143,628)(144,627)(145,626)(146,625)(147,624)(148,623)(149,622)(150,621)(151,620)(152,619)(153,618)(154,617)(155,616)(156,615)(157,614)(158,613)(159,612)(160,611)(161,610)(162,609)(163,608)(164,607)(165,606)(166,605)(167,604)(168,603)(169,602)(170,601)(171,600)(172,599)(173,598)(174,597)(175,596)(176,595)(177,594)(178,593)(179,592)(180,591)(181,590)(182,589)(183,588)(184,587)(185,586)(186,585)(187,584)(188,583)(189,582)(190,581)(191,580)(192,579)(193,578)(194,577)(195,576)(196,575)(197,574)(198,573)(199,572)(200,571)(201,570)(202,569)(203,568)(204,567)(205,566)(206,565)(207,564)(208,563)(209,562)(210,561)(211,560)(212,559)(213,558)(214,557)(215,556)(216,555)(217,554)(218,553)(219,552)(220,551)(221,550)(222,549)(223,548)(224,547)(225,546)(226,545)(227,544)(228,543)(229,542)(230,541)(231,540)(232,539)(233,538)(234,537)(235,536)(236,535)(237,534)(238,533)(239,532)(240,531)(241,530)(242,529)(243,528)(244,527)(245,526)(246,525)(247,524)(248,523)(249,522)(250,521)(251,520)(252,519)(253,518)(254,517)(255,516)(256,515)(257,514)(258,513)(259,512)(260,511)(261,510)(262,509)(263,508)(264,507)(265,506)(266,505)(267,504)(268,503)(269,502)(270,501)(271,500)(272,499)(273,498)(274,497)(275,496)(276,495)(277,494)(278,493)(279,492)(280,491)(281,490)(282,489)(283,488)(284,487)(285,486)(286,485)(287,484)(288,483)(289,482)(290,481)(291,480)(292,479)(293,478)(294,477)(295,476)(296,475)(297,474)(298,473)(299,472)(300,471)(301,470)(302,469)(303,468)(304,467)(305,466)(306,465)(307,464)(308,463)(309,462)(310,461)(311,460)(312,459)(313,458)(314,457)(315,456)(316,455)(317,454)(318,453)(319,452)(320,451)(321,450)(322,449)(323,448)(324,447)(325,446)(326,445)(327,444)(328,443)(329,442)(330,441)(331,440)(332,439)(333,438)(334,437)(335,436)(336,435)(337,434)(338,433)(339,432)(340,431)(341,430)(342,429)(343,428)(344,427)(345,426)(346,425)(347,424)(348,423)(349,422)(350,421)(351,420)(352,419)(353,418)(354,417)(355,416)(356,415)(357,414)(358,413)(359,412)(360,411)(361,410)(362,409)(363,408)(364,407)(365,406)(366,405)(367,404)(368,403)(369,402)(370,401)(371,400)(372,399)(373,398)(374,397)(375,396)(376,395)(377,394)(378,393)(379,392)(380,391)(381,390)(382,389)(383,388)(384,387)(385,386)

Polycyclic presentation:

2,11,739,2|a^2=d,b^a=b^10,c^a=c^738

Conjugacy classes & character table

8132 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₂₅₈
maximal normal subgroup of	Dic₁₆₂₅₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32516, 4);

Dic8129 #45654170