D₇₂₉₉ #28410263

order	14598
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₂₉₉
unambiguous semidirect product	C₉⋊D₈₁₁
unambiguous semidirect product	C₈₁₁⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₈₁₁ ⋊ C₂
Sylow	C₈₁₁ ⋊ D₉
Sylow	C₉ ⋊ D₈₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₈₁₁	C₂
Frattini	C₃	D₂₄₃₃
socle	C₃×C₈₁₁	D₃
LCS	G ▷ C₉×C₈₁₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 820
transitive degree	μ_t(G) = 7299

A minimal faithful permutation representation — 2 generators on 820 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820);(2,9)(3,8)(4,7)(5,6)(11,820)(12,819)(13,818)(14,817)(15,816)(16,815)(17,814)(18,813)(19,812)(20,811)(21,810)(22,809)(23,808)(24,807)(25,806)(26,805)(27,804)(28,803)(29,802)(30,801)(31,800)(32,799)(33,798)(34,797)(35,796)(36,795)(37,794)(38,793)(39,792)(40,791)(41,790)(42,789)(43,788)(44,787)(45,786)(46,785)(47,784)(48,783)(49,782)(50,781)(51,780)(52,779)(53,778)(54,777)(55,776)(56,775)(57,774)(58,773)(59,772)(60,771)(61,770)(62,769)(63,768)(64,767)(65,766)(66,765)(67,764)(68,763)(69,762)(70,761)(71,760)(72,759)(73,758)(74,757)(75,756)(76,755)(77,754)(78,753)(79,752)(80,751)(81,750)(82,749)(83,748)(84,747)(85,746)(86,745)(87,744)(88,743)(89,742)(90,741)(91,740)(92,739)(93,738)(94,737)(95,736)(96,735)(97,734)(98,733)(99,732)(100,731)(101,730)(102,729)(103,728)(104,727)(105,726)(106,725)(107,724)(108,723)(109,722)(110,721)(111,720)(112,719)(113,718)(114,717)(115,716)(116,715)(117,714)(118,713)(119,712)(120,711)(121,710)(122,709)(123,708)(124,707)(125,706)(126,705)(127,704)(128,703)(129,702)(130,701)(131,700)(132,699)(133,698)(134,697)(135,696)(136,695)(137,694)(138,693)(139,692)(140,691)(141,690)(142,689)(143,688)(144,687)(145,686)(146,685)(147,684)(148,683)(149,682)(150,681)(151,680)(152,679)(153,678)(154,677)(155,676)(156,675)(157,674)(158,673)(159,672)(160,671)(161,670)(162,669)(163,668)(164,667)(165,666)(166,665)(167,664)(168,663)(169,662)(170,661)(171,660)(172,659)(173,658)(174,657)(175,656)(176,655)(177,654)(178,653)(179,652)(180,651)(181,650)(182,649)(183,648)(184,647)(185,646)(186,645)(187,644)(188,643)(189,642)(190,641)(191,640)(192,639)(193,638)(194,637)(195,636)(196,635)(197,634)(198,633)(199,632)(200,631)(201,630)(202,629)(203,628)(204,627)(205,626)(206,625)(207,624)(208,623)(209,622)(210,621)(211,620)(212,619)(213,618)(214,617)(215,616)(216,615)(217,614)(218,613)(219,612)(220,611)(221,610)(222,609)(223,608)(224,607)(225,606)(226,605)(227,604)(228,603)(229,602)(230,601)(231,600)(232,599)(233,598)(234,597)(235,596)(236,595)(237,594)(238,593)(239,592)(240,591)(241,590)(242,589)(243,588)(244,587)(245,586)(246,585)(247,584)(248,583)(249,582)(250,581)(251,580)(252,579)(253,578)(254,577)(255,576)(256,575)(257,574)(258,573)(259,572)(260,571)(261,570)(262,569)(263,568)(264,567)(265,566)(266,565)(267,564)(268,563)(269,562)(270,561)(271,560)(272,559)(273,558)(274,557)(275,556)(276,555)(277,554)(278,553)(279,552)(280,551)(281,550)(282,549)(283,548)(284,547)(285,546)(286,545)(287,544)(288,543)(289,542)(290,541)(291,540)(292,539)(293,538)(294,537)(295,536)(296,535)(297,534)(298,533)(299,532)(300,531)(301,530)(302,529)(303,528)(304,527)(305,526)(306,525)(307,524)(308,523)(309,522)(310,521)(311,520)(312,519)(313,518)(314,517)(315,516)(316,515)(317,514)(318,513)(319,512)(320,511)(321,510)(322,509)(323,508)(324,507)(325,506)(326,505)(327,504)(328,503)(329,502)(330,501)(331,500)(332,499)(333,498)(334,497)(335,496)(336,495)(337,494)(338,493)(339,492)(340,491)(341,490)(342,489)(343,488)(344,487)(345,486)(346,485)(347,484)(348,483)(349,482)(350,481)(351,480)(352,479)(353,478)(354,477)(355,476)(356,475)(357,474)(358,473)(359,472)(360,471)(361,470)(362,469)(363,468)(364,467)(365,466)(366,465)(367,464)(368,463)(369,462)(370,461)(371,460)(372,459)(373,458)(374,457)(375,456)(376,455)(377,454)(378,453)(379,452)(380,451)(381,450)(382,449)(383,448)(384,447)(385,446)(386,445)(387,444)(388,443)(389,442)(390,441)(391,440)(392,439)(393,438)(394,437)(395,436)(396,435)(397,434)(398,433)(399,432)(400,431)(401,430)(402,429)(403,428)(404,427)(405,426)(406,425)(407,424)(408,423)(409,422)(410,421)(411,420)(412,419)(413,418)(414,417)(415,416)

Polycyclic presentation:

2,3,811,3|b^3=d^2,b^a=b^2d,c^a=c^810,d^a=d^2

Conjugacy classes & character table

3651 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₈₁₁, C₂×D₇₂₉₉, D₂₁₈₉₇
maximal normal subgroup of	D₉×D₈₁₁, C₂×D₇₂₉₉
maximal quotient of	Dic₇₂₉₉, C₂×D₇₂₉₉, D₂₁₈₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14598, 6);

D7299 #28410263