D₈₁₂₄ #72213136

order	16248
automorphism	Aut(G) order 21967296
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₁₂₄

Splittings

Fitting	C₃×C₄×C₆₇₇ ⋊ C₂
Hall	C₃×C₆₇₇ ⋊ D₄
Sylow	C₆₇₇ ⋊ D₁₂
Sylow	C₃ ⋊ D₂₇₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₃₁
commutator, socle, cosocle	C₂×C₃×C₆₇₇	C₂²
Fitting	C₃×C₄×C₆₇₇	C₂
hypercenter	C₄	D₂₀₃₁
hypocenter	C₃×C₆₇₇	D₄
LCS	G ▷ C₂×C₃×C₆₇₇ ▷ C₃×C₆₇₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 684
transitive degree	μ_t(G) = 8124

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

Polycyclic presentation:

2^3,3,677|b^2=c,b^a=bc,d^a=d^2,e^a=e^676

Conjugacy classes & character table

4065 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₂₄₈, D₂₄₃₇₂
maximal normal subgroup of	D₁₆₂₄₈
maximal quotient of	Dic₈₁₂₄, D₁₆₂₄₈, D₂₄₃₇₂

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, 677]);;
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.5^676);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8124 #72213136