D₁₃₅₁₂ #72235506

order	27024
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₁₂

Splittings

Fitting	C₃×C₈×C₅₆₃ ⋊ C₂
Hall	C₃×C₅₆₃ ⋊ D₈
Sylow	C₅₆₃ ⋊ D₂₄
Sylow	C₃ ⋊ D₄₅₀₄

Subgroups

	subgroup	G/H
center	C₂	D₆₇₅₆
commutator, cosocle	C₃×C₄×C₅₆₃	C₂²
Frattini	C₄	C₂×D₁₆₈₉
Fitting	C₃×C₈×C₅₆₃	C₂
socle	C₂×C₃×C₅₆₃	D₄
hypercenter	C₈	D₁₆₈₉
hypocenter	C₃×C₅₆₃	D₈
LCS	G ▷ C₃×C₄×C₅₆₃ ▷ C₂×C₃×C₅₆₃ ▷ C₃×C₅₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 574
transitive degree	μ_t(G) = 13512

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

Polycyclic presentation:

2^4,3,563|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^562

Conjugacy classes & character table

6759 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₀₂₄
maximal normal subgroup of	D₂₇₀₂₄
maximal quotient of	Dic₁₃₅₁₂, D₂₇₀₂₄

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, 563]);;
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.6^562);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13512 #72235506