Dic₇₉₀₉ #44539496

order	31636
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₉₀₉
unambiguous semidirect product	C₁₁⋊Dic₇₁₉
unambiguous semidirect product	C₇₁₉⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₇₁₉ ⋊ C₄
Sylow	C₇₁₉ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₇₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₉₀₉
commutator, hypocenter	C₁₁×C₇₁₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₇₁₉	C₂
LCS	G ▷ C₁₁×C₇₁₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 734
transitive degree	μ_t(G) = 31636

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

Polycyclic presentation:

2,11,719,2|a^2=d,b^a=b^10,c^a=c^718

Conjugacy classes & character table

7912 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₈₁₈
maximal normal subgroup of	Dic₁₅₈₁₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31636, 4);

Dic7909 #44539496