Dic₈₀₆₃ #44567485

order	32252
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₆₃
unambiguous semidirect product	C₁₁⋊Dic₇₃₃

Splittings

derived, Hall	C₁₁×C₇₃₃ ⋊ C₄
Sylow	C₇₃₃ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₇₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₆₃
commutator, hypocenter	C₁₁×C₇₃₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₇₃₃	C₂
LCS	G ▷ C₁₁×C₇₃₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 748
transitive degree	μ_t(G) = 32252

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

Polycyclic presentation:

2,11,733,2|a^2=d,b^a=b^10,c^a=c^732

Conjugacy classes & character table

8066 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₁₂₆
maximal normal subgroup of	Dic₁₆₁₂₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32252, 4);

Dic8063 #44567485