D₁₄₄₂₄ #72235751

order	28848
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₄₂₄

Splittings

Fitting	C₃×C₈×C₆₀₁ ⋊ C₂
Hall	C₃×C₆₀₁ ⋊ D₈
Sylow	C₆₀₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₄₈₀₈

Subgroups

	subgroup	G/H
center	C₂	D₇₂₁₂
commutator, cosocle	C₃×C₄×C₆₀₁	C₂²
Frattini	C₄	C₂×D₁₈₀₃
Fitting	C₃×C₈×C₆₀₁	C₂
socle	C₂×C₃×C₆₀₁	D₄
hypercenter	C₈	D₁₈₀₃
hypocenter	C₃×C₆₀₁	D₈
LCS	G ▷ C₃×C₄×C₆₀₁ ▷ C₂×C₃×C₆₀₁ ▷ C₃×C₆₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 612
transitive degree	μ_t(G) = 14424

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

Polycyclic presentation:

2^4,3,601|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^600

Conjugacy classes & character table

7215 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₈₄₈
maximal normal subgroup of	D₂₈₈₄₈
maximal quotient of	Dic₁₄₄₂₄, D₂₈₈₄₈

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, 601]);;
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.6^600);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14424 #72235751