Dic₆₁₂₇ #37587745

order	24508
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₂₇
unambiguous semidirect product	C₁₁⋊Dic₅₅₇

Splittings

derived, Hall	C₁₁×C₅₅₇ ⋊ C₄
Sylow	C₅₅₇ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₅₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁₂₇
commutator, hypocenter	C₁₁×C₅₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₅₅₇	C₂
LCS	G ▷ C₁₁×C₅₅₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 572
transitive degree	μ_t(G) = 24508

A minimal faithful permutation representation — 2 generators on 572 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,572)(18,571)(19,570)(20,569)(21,568)(22,567)(23,566)(24,565)(25,564)(26,563)(27,562)(28,561)(29,560)(30,559)(31,558)(32,557)(33,556)(34,555)(35,554)(36,553)(37,552)(38,551)(39,550)(40,549)(41,548)(42,547)(43,546)(44,545)(45,544)(46,543)(47,542)(48,541)(49,540)(50,539)(51,538)(52,537)(53,536)(54,535)(55,534)(56,533)(57,532)(58,531)(59,530)(60,529)(61,528)(62,527)(63,526)(64,525)(65,524)(66,523)(67,522)(68,521)(69,520)(70,519)(71,518)(72,517)(73,516)(74,515)(75,514)(76,513)(77,512)(78,511)(79,510)(80,509)(81,508)(82,507)(83,506)(84,505)(85,504)(86,503)(87,502)(88,501)(89,500)(90,499)(91,498)(92,497)(93,496)(94,495)(95,494)(96,493)(97,492)(98,491)(99,490)(100,489)(101,488)(102,487)(103,486)(104,485)(105,484)(106,483)(107,482)(108,481)(109,480)(110,479)(111,478)(112,477)(113,476)(114,475)(115,474)(116,473)(117,472)(118,471)(119,470)(120,469)(121,468)(122,467)(123,466)(124,465)(125,464)(126,463)(127,462)(128,461)(129,460)(130,459)(131,458)(132,457)(133,456)(134,455)(135,454)(136,453)(137,452)(138,451)(139,450)(140,449)(141,448)(142,447)(143,446)(144,445)(145,444)(146,443)(147,442)(148,441)(149,440)(150,439)(151,438)(152,437)(153,436)(154,435)(155,434)(156,433)(157,432)(158,431)(159,430)(160,429)(161,428)(162,427)(163,426)(164,425)(165,424)(166,423)(167,422)(168,421)(169,420)(170,419)(171,418)(172,417)(173,416)(174,415)(175,414)(176,413)(177,412)(178,411)(179,410)(180,409)(181,408)(182,407)(183,406)(184,405)(185,404)(186,403)(187,402)(188,401)(189,400)(190,399)(191,398)(192,397)(193,396)(194,395)(195,394)(196,393)(197,392)(198,391)(199,390)(200,389)(201,388)(202,387)(203,386)(204,385)(205,384)(206,383)(207,382)(208,381)(209,380)(210,379)(211,378)(212,377)(213,376)(214,375)(215,374)(216,373)(217,372)(218,371)(219,370)(220,369)(221,368)(222,367)(223,366)(224,365)(225,364)(226,363)(227,362)(228,361)(229,360)(230,359)(231,358)(232,357)(233,356)(234,355)(235,354)(236,353)(237,352)(238,351)(239,350)(240,349)(241,348)(242,347)(243,346)(244,345)(245,344)(246,343)(247,342)(248,341)(249,340)(250,339)(251,338)(252,337)(253,336)(254,335)(255,334)(256,333)(257,332)(258,331)(259,330)(260,329)(261,328)(262,327)(263,326)(264,325)(265,324)(266,323)(267,322)(268,321)(269,320)(270,319)(271,318)(272,317)(273,316)(274,315)(275,314)(276,313)(277,312)(278,311)(279,310)(280,309)(281,308)(282,307)(283,306)(284,305)(285,304)(286,303)(287,302)(288,301)(289,300)(290,299)(291,298)(292,297)(293,296)(294,295)

Polycyclic presentation:

2,11,557,2|a^2=d,b^a=b^10,c^a=c^556

Conjugacy classes & character table

6130 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₂₅₄
maximal normal subgroup of	Dic₁₂₂₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24508, 4);

Dic6127 #37587745