Dic₇₇₉₉ #44530058

order	31196
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₇₉₉
unambiguous semidirect product	C₁₁⋊Dic₇₀₉

Splittings

derived, Hall	C₁₁×C₇₀₉ ⋊ C₄
Sylow	C₇₀₉ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₇₀₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₇₉₉
commutator, hypocenter	C₁₁×C₇₀₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₇₀₉	C₂
LCS	G ▷ C₁₁×C₇₀₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 724
transitive degree	μ_t(G) = 31196

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

Polycyclic presentation:

2,11,709,2|a^2=d,b^a=b^10,c^a=c^708

Conjugacy classes & character table

7802 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₅₉₈
maximal normal subgroup of	Dic₁₅₅₉₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31196, 4);

Dic7799 #44530058