D₁₅₆₇₂ #72272302

order	31344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₆₇₂

Splittings

Fitting	C₃×C₈×C₆₅₃ ⋊ C₂
Hall	C₃×C₆₅₃ ⋊ D₈
Sylow	C₆₅₃ ⋊ D₂₄
Sylow	C₃ ⋊ D₅₂₂₄

Subgroups

	subgroup	G/H
center	C₂	D₇₈₃₆
commutator, cosocle	C₃×C₄×C₆₅₃	C₂²
Frattini	C₄	C₂×D₁₉₅₉
Fitting	C₃×C₈×C₆₅₃	C₂
socle	C₂×C₃×C₆₅₃	D₄
hypercenter	C₈	D₁₉₅₉
hypocenter	C₃×C₆₅₃	D₈
LCS	G ▷ C₃×C₄×C₆₅₃ ▷ C₂×C₃×C₆₅₃ ▷ C₃×C₆₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 664
transitive degree	μ_t(G) = 15672

A minimal faithful permutation representation — 2 generators on 664 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);(2,8)(3,7)(4,6)(10,11)(13,664)(14,663)(15,662)(16,661)(17,660)(18,659)(19,658)(20,657)(21,656)(22,655)(23,654)(24,653)(25,652)(26,651)(27,650)(28,649)(29,648)(30,647)(31,646)(32,645)(33,644)(34,643)(35,642)(36,641)(37,640)(38,639)(39,638)(40,637)(41,636)(42,635)(43,634)(44,633)(45,632)(46,631)(47,630)(48,629)(49,628)(50,627)(51,626)(52,625)(53,624)(54,623)(55,622)(56,621)(57,620)(58,619)(59,618)(60,617)(61,616)(62,615)(63,614)(64,613)(65,612)(66,611)(67,610)(68,609)(69,608)(70,607)(71,606)(72,605)(73,604)(74,603)(75,602)(76,601)(77,600)(78,599)(79,598)(80,597)(81,596)(82,595)(83,594)(84,593)(85,592)(86,591)(87,590)(88,589)(89,588)(90,587)(91,586)(92,585)(93,584)(94,583)(95,582)(96,581)(97,580)(98,579)(99,578)(100,577)(101,576)(102,575)(103,574)(104,573)(105,572)(106,571)(107,570)(108,569)(109,568)(110,567)(111,566)(112,565)(113,564)(114,563)(115,562)(116,561)(117,560)(118,559)(119,558)(120,557)(121,556)(122,555)(123,554)(124,553)(125,552)(126,551)(127,550)(128,549)(129,548)(130,547)(131,546)(132,545)(133,544)(134,543)(135,542)(136,541)(137,540)(138,539)(139,538)(140,537)(141,536)(142,535)(143,534)(144,533)(145,532)(146,531)(147,530)(148,529)(149,528)(150,527)(151,526)(152,525)(153,524)(154,523)(155,522)(156,521)(157,520)(158,519)(159,518)(160,517)(161,516)(162,515)(163,514)(164,513)(165,512)(166,511)(167,510)(168,509)(169,508)(170,507)(171,506)(172,505)(173,504)(174,503)(175,502)(176,501)(177,500)(178,499)(179,498)(180,497)(181,496)(182,495)(183,494)(184,493)(185,492)(186,491)(187,490)(188,489)(189,488)(190,487)(191,486)(192,485)(193,484)(194,483)(195,482)(196,481)(197,480)(198,479)(199,478)(200,477)(201,476)(202,475)(203,474)(204,473)(205,472)(206,471)(207,470)(208,469)(209,468)(210,467)(211,466)(212,465)(213,464)(214,463)(215,462)(216,461)(217,460)(218,459)(219,458)(220,457)(221,456)(222,455)(223,454)(224,453)(225,452)(226,451)(227,450)(228,449)(229,448)(230,447)(231,446)(232,445)(233,444)(234,443)(235,442)(236,441)(237,440)(238,439)(239,438)(240,437)(241,436)(242,435)(243,434)(244,433)(245,432)(246,431)(247,430)(248,429)(249,428)(250,427)(251,426)(252,425)(253,424)(254,423)(255,422)(256,421)(257,420)(258,419)(259,418)(260,417)(261,416)(262,415)(263,414)(264,413)(265,412)(266,411)(267,410)(268,409)(269,408)(270,407)(271,406)(272,405)(273,404)(274,403)(275,402)(276,401)(277,400)(278,399)(279,398)(280,397)(281,396)(282,395)(283,394)(284,393)(285,392)(286,391)(287,390)(288,389)(289,388)(290,387)(291,386)(292,385)(293,384)(294,383)(295,382)(296,381)(297,380)(298,379)(299,378)(300,377)(301,376)(302,375)(303,374)(304,373)(305,372)(306,371)(307,370)(308,369)(309,368)(310,367)(311,366)(312,365)(313,364)(314,363)(315,362)(316,361)(317,360)(318,359)(319,358)(320,357)(321,356)(322,355)(323,354)(324,353)(325,352)(326,351)(327,350)(328,349)(329,348)(330,347)(331,346)(332,345)(333,344)(334,343)(335,342)(336,341)(337,340)(338,339)

Polycyclic presentation:

2^4,3,653|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^652

Conjugacy classes & character table

7839 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁₃₄₄
maximal normal subgroup of	D₃₁₃₄₄
maximal quotient of	Dic₁₅₆₇₂, D₃₁₃₄₄

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 653]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^652);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15672 #72272302