D₈₇₉₆ #72213675

order	17592
automorphism	Aut(G) order 25754688
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₉₆

Splittings

Fitting	C₃×C₄×C₇₃₃ ⋊ C₂
Hall	C₃×C₇₃₃ ⋊ D₄
Sylow	C₇₃₃ ⋊ D₁₂
Sylow	C₃ ⋊ D₂₉₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₉₉
commutator, socle, cosocle	C₂×C₃×C₇₃₃	C₂²
Fitting	C₃×C₄×C₇₃₃	C₂
hypercenter	C₄	D₂₁₉₉
hypocenter	C₃×C₇₃₃	D₄
LCS	G ▷ C₂×C₃×C₇₃₃ ▷ C₃×C₇₃₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 740
transitive degree	μ_t(G) = 8796

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

Polycyclic presentation:

2^3,3,733|b^2=c,b^a=bc,d^a=d^2,e^a=e^732

Conjugacy classes & character table

4401 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₅₉₂, D₂₆₃₈₈
maximal normal subgroup of	D₁₇₅₉₂
maximal quotient of	Dic₈₇₉₆, D₁₇₅₉₂, D₂₆₃₈₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 733]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^732);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8796 #72213675