D₁₁₇₄₇ #37555040

order	23494
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₄₇
unambiguous semidirect product	C₁₇⋊D₆₉₁
unambiguous semidirect product	C₆₉₁⋊D₁₇

Splittings

Fitting, socle, derived, Hall	C₁₇×C₆₉₁ ⋊ C₂
Sylow	C₆₉₁ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₆₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₇×C₆₉₁	C₂
LCS	G ▷ C₁₇×C₆₉₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 708
transitive degree	μ_t(G) = 11747

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

Polycyclic presentation:

2,17,691|b^a=b^16,c^a=c^690

Conjugacy classes & character table

5875 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23494, 3);

D11747 #37555040