D₁₆₆₀₉ #45684690

order	33218
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₆₀₉
unambiguous semidirect product	C₁₇⋊D₉₇₇
unambiguous semidirect product	C₉₇₇⋊D₁₇

Splittings

Fitting, socle, derived, Hall	C₁₇×C₉₇₇ ⋊ C₂
Sylow	C₉₇₇ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₉₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₇×C₉₇₇	C₂
LCS	G ▷ C₁₇×C₉₇₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 994
transitive degree	μ_t(G) = 16609

A minimal faithful permutation representation — 2 generators on 994 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994);(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,994)(20,993)(21,992)(22,991)(23,990)(24,989)(25,988)(26,987)(27,986)(28,985)(29,984)(30,983)(31,982)(32,981)(33,980)(34,979)(35,978)(36,977)(37,976)(38,975)(39,974)(40,973)(41,972)(42,971)(43,970)(44,969)(45,968)(46,967)(47,966)(48,965)(49,964)(50,963)(51,962)(52,961)(53,960)(54,959)(55,958)(56,957)(57,956)(58,955)(59,954)(60,953)(61,952)(62,951)(63,950)(64,949)(65,948)(66,947)(67,946)(68,945)(69,944)(70,943)(71,942)(72,941)(73,940)(74,939)(75,938)(76,937)(77,936)(78,935)(79,934)(80,933)(81,932)(82,931)(83,930)(84,929)(85,928)(86,927)(87,926)(88,925)(89,924)(90,923)(91,922)(92,921)(93,920)(94,919)(95,918)(96,917)(97,916)(98,915)(99,914)(100,913)(101,912)(102,911)(103,910)(104,909)(105,908)(106,907)(107,906)(108,905)(109,904)(110,903)(111,902)(112,901)(113,900)(114,899)(115,898)(116,897)(117,896)(118,895)(119,894)(120,893)(121,892)(122,891)(123,890)(124,889)(125,888)(126,887)(127,886)(128,885)(129,884)(130,883)(131,882)(132,881)(133,880)(134,879)(135,878)(136,877)(137,876)(138,875)(139,874)(140,873)(141,872)(142,871)(143,870)(144,869)(145,868)(146,867)(147,866)(148,865)(149,864)(150,863)(151,862)(152,861)(153,860)(154,859)(155,858)(156,857)(157,856)(158,855)(159,854)(160,853)(161,852)(162,851)(163,850)(164,849)(165,848)(166,847)(167,846)(168,845)(169,844)(170,843)(171,842)(172,841)(173,840)(174,839)(175,838)(176,837)(177,836)(178,835)(179,834)(180,833)(181,832)(182,831)(183,830)(184,829)(185,828)(186,827)(187,826)(188,825)(189,824)(190,823)(191,822)(192,821)(193,820)(194,819)(195,818)(196,817)(197,816)(198,815)(199,814)(200,813)(201,812)(202,811)(203,810)(204,809)(205,808)(206,807)(207,806)(208,805)(209,804)(210,803)(211,802)(212,801)(213,800)(214,799)(215,798)(216,797)(217,796)(218,795)(219,794)(220,793)(221,792)(222,791)(223,790)(224,789)(225,788)(226,787)(227,786)(228,785)(229,784)(230,783)(231,782)(232,781)(233,780)(234,779)(235,778)(236,777)(237,776)(238,775)(239,774)(240,773)(241,772)(242,771)(243,770)(244,769)(245,768)(246,767)(247,766)(248,765)(249,764)(250,763)(251,762)(252,761)(253,760)(254,759)(255,758)(256,757)(257,756)(258,755)(259,754)(260,753)(261,752)(262,751)(263,750)(264,749)(265,748)(266,747)(267,746)(268,745)(269,744)(270,743)(271,742)(272,741)(273,740)(274,739)(275,738)(276,737)(277,736)(278,735)(279,734)(280,733)(281,732)(282,731)(283,730)(284,729)(285,728)(286,727)(287,726)(288,725)(289,724)(290,723)(291,722)(292,721)(293,720)(294,719)(295,718)(296,717)(297,716)(298,715)(299,714)(300,713)(301,712)(302,711)(303,710)(304,709)(305,708)(306,707)(307,706)(308,705)(309,704)(310,703)(311,702)(312,701)(313,700)(314,699)(315,698)(316,697)(317,696)(318,695)(319,694)(320,693)(321,692)(322,691)(323,690)(324,689)(325,688)(326,687)(327,686)(328,685)(329,684)(330,683)(331,682)(332,681)(333,680)(334,679)(335,678)(336,677)(337,676)(338,675)(339,674)(340,673)(341,672)(342,671)(343,670)(344,669)(345,668)(346,667)(347,666)(348,665)(349,664)(350,663)(351,662)(352,661)(353,660)(354,659)(355,658)(356,657)(357,656)(358,655)(359,654)(360,653)(361,652)(362,651)(363,650)(364,649)(365,648)(366,647)(367,646)(368,645)(369,644)(370,643)(371,642)(372,641)(373,640)(374,639)(375,638)(376,637)(377,636)(378,635)(379,634)(380,633)(381,632)(382,631)(383,630)(384,629)(385,628)(386,627)(387,626)(388,625)(389,624)(390,623)(391,622)(392,621)(393,620)(394,619)(395,618)(396,617)(397,616)(398,615)(399,614)(400,613)(401,612)(402,611)(403,610)(404,609)(405,608)(406,607)(407,606)(408,605)(409,604)(410,603)(411,602)(412,601)(413,600)(414,599)(415,598)(416,597)(417,596)(418,595)(419,594)(420,593)(421,592)(422,591)(423,590)(424,589)(425,588)(426,587)(427,586)(428,585)(429,584)(430,583)(431,582)(432,581)(433,580)(434,579)(435,578)(436,577)(437,576)(438,575)(439,574)(440,573)(441,572)(442,571)(443,570)(444,569)(445,568)(446,567)(447,566)(448,565)(449,564)(450,563)(451,562)(452,561)(453,560)(454,559)(455,558)(456,557)(457,556)(458,555)(459,554)(460,553)(461,552)(462,551)(463,550)(464,549)(465,548)(466,547)(467,546)(468,545)(469,544)(470,543)(471,542)(472,541)(473,540)(474,539)(475,538)(476,537)(477,536)(478,535)(479,534)(480,533)(481,532)(482,531)(483,530)(484,529)(485,528)(486,527)(487,526)(488,525)(489,524)(490,523)(491,522)(492,521)(493,520)(494,519)(495,518)(496,517)(497,516)(498,515)(499,514)(500,513)(501,512)(502,511)(503,510)(504,509)(505,508)(506,507)

Polycyclic presentation:

2,17,977|b^a=b^16,c^a=c^976

Conjugacy classes & character table

8306 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33218, 3);

D16609 #45684690