Dic₆₂₂₇ #38735331

order	24908
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₂₂₇
unambiguous semidirect product	C₄₇₉⋊Dic₁₃

Splittings

derived, Hall	C₁₃×C₄₇₉ ⋊ C₄
Sylow	C₄₇₉ ⋊ Dic₁₃
Sylow	C₁₃ ⋊ Dic₄₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₂₂₇
commutator, hypocenter	C₁₃×C₄₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₄₇₉	C₂
LCS	G ▷ C₁₃×C₄₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 496
transitive degree	μ_t(G) = 24908

A minimal faithful permutation representation — 2 generators on 496 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,496)(20,495)(21,494)(22,493)(23,492)(24,491)(25,490)(26,489)(27,488)(28,487)(29,486)(30,485)(31,484)(32,483)(33,482)(34,481)(35,480)(36,479)(37,478)(38,477)(39,476)(40,475)(41,474)(42,473)(43,472)(44,471)(45,470)(46,469)(47,468)(48,467)(49,466)(50,465)(51,464)(52,463)(53,462)(54,461)(55,460)(56,459)(57,458)(58,457)(59,456)(60,455)(61,454)(62,453)(63,452)(64,451)(65,450)(66,449)(67,448)(68,447)(69,446)(70,445)(71,444)(72,443)(73,442)(74,441)(75,440)(76,439)(77,438)(78,437)(79,436)(80,435)(81,434)(82,433)(83,432)(84,431)(85,430)(86,429)(87,428)(88,427)(89,426)(90,425)(91,424)(92,423)(93,422)(94,421)(95,420)(96,419)(97,418)(98,417)(99,416)(100,415)(101,414)(102,413)(103,412)(104,411)(105,410)(106,409)(107,408)(108,407)(109,406)(110,405)(111,404)(112,403)(113,402)(114,401)(115,400)(116,399)(117,398)(118,397)(119,396)(120,395)(121,394)(122,393)(123,392)(124,391)(125,390)(126,389)(127,388)(128,387)(129,386)(130,385)(131,384)(132,383)(133,382)(134,381)(135,380)(136,379)(137,378)(138,377)(139,376)(140,375)(141,374)(142,373)(143,372)(144,371)(145,370)(146,369)(147,368)(148,367)(149,366)(150,365)(151,364)(152,363)(153,362)(154,361)(155,360)(156,359)(157,358)(158,357)(159,356)(160,355)(161,354)(162,353)(163,352)(164,351)(165,350)(166,349)(167,348)(168,347)(169,346)(170,345)(171,344)(172,343)(173,342)(174,341)(175,340)(176,339)(177,338)(178,337)(179,336)(180,335)(181,334)(182,333)(183,332)(184,331)(185,330)(186,329)(187,328)(188,327)(189,326)(190,325)(191,324)(192,323)(193,322)(194,321)(195,320)(196,319)(197,318)(198,317)(199,316)(200,315)(201,314)(202,313)(203,312)(204,311)(205,310)(206,309)(207,308)(208,307)(209,306)(210,305)(211,304)(212,303)(213,302)(214,301)(215,300)(216,299)(217,298)(218,297)(219,296)(220,295)(221,294)(222,293)(223,292)(224,291)(225,290)(226,289)(227,288)(228,287)(229,286)(230,285)(231,284)(232,283)(233,282)(234,281)(235,280)(236,279)(237,278)(238,277)(239,276)(240,275)(241,274)(242,273)(243,272)(244,271)(245,270)(246,269)(247,268)(248,267)(249,266)(250,265)(251,264)(252,263)(253,262)(254,261)(255,260)(256,259)(257,258)

Polycyclic presentation:

2,13,479,2|a^2=d,b^a=b^12,c^a=c^478

Conjugacy classes & character table

6230 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₄₅₄
maximal normal subgroup of	Dic₁₂₄₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24908, 4);

Dic6227 #38735331