Dic₆₇₄₃ #41004222

order	26972
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₇₄₃
unambiguous semidirect product	C₁₁⋊Dic₆₁₃

Splittings

derived, Hall	C₁₁×C₆₁₃ ⋊ C₄
Sylow	C₆₁₃ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₆₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₇₄₃
commutator, hypocenter	C₁₁×C₆₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₆₁₃	C₂
LCS	G ▷ C₁₁×C₆₁₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 628
transitive degree	μ_t(G) = 26972

A minimal faithful permutation representation — 2 generators on 628 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,628)(18,627)(19,626)(20,625)(21,624)(22,623)(23,622)(24,621)(25,620)(26,619)(27,618)(28,617)(29,616)(30,615)(31,614)(32,613)(33,612)(34,611)(35,610)(36,609)(37,608)(38,607)(39,606)(40,605)(41,604)(42,603)(43,602)(44,601)(45,600)(46,599)(47,598)(48,597)(49,596)(50,595)(51,594)(52,593)(53,592)(54,591)(55,590)(56,589)(57,588)(58,587)(59,586)(60,585)(61,584)(62,583)(63,582)(64,581)(65,580)(66,579)(67,578)(68,577)(69,576)(70,575)(71,574)(72,573)(73,572)(74,571)(75,570)(76,569)(77,568)(78,567)(79,566)(80,565)(81,564)(82,563)(83,562)(84,561)(85,560)(86,559)(87,558)(88,557)(89,556)(90,555)(91,554)(92,553)(93,552)(94,551)(95,550)(96,549)(97,548)(98,547)(99,546)(100,545)(101,544)(102,543)(103,542)(104,541)(105,540)(106,539)(107,538)(108,537)(109,536)(110,535)(111,534)(112,533)(113,532)(114,531)(115,530)(116,529)(117,528)(118,527)(119,526)(120,525)(121,524)(122,523)(123,522)(124,521)(125,520)(126,519)(127,518)(128,517)(129,516)(130,515)(131,514)(132,513)(133,512)(134,511)(135,510)(136,509)(137,508)(138,507)(139,506)(140,505)(141,504)(142,503)(143,502)(144,501)(145,500)(146,499)(147,498)(148,497)(149,496)(150,495)(151,494)(152,493)(153,492)(154,491)(155,490)(156,489)(157,488)(158,487)(159,486)(160,485)(161,484)(162,483)(163,482)(164,481)(165,480)(166,479)(167,478)(168,477)(169,476)(170,475)(171,474)(172,473)(173,472)(174,471)(175,470)(176,469)(177,468)(178,467)(179,466)(180,465)(181,464)(182,463)(183,462)(184,461)(185,460)(186,459)(187,458)(188,457)(189,456)(190,455)(191,454)(192,453)(193,452)(194,451)(195,450)(196,449)(197,448)(198,447)(199,446)(200,445)(201,444)(202,443)(203,442)(204,441)(205,440)(206,439)(207,438)(208,437)(209,436)(210,435)(211,434)(212,433)(213,432)(214,431)(215,430)(216,429)(217,428)(218,427)(219,426)(220,425)(221,424)(222,423)(223,422)(224,421)(225,420)(226,419)(227,418)(228,417)(229,416)(230,415)(231,414)(232,413)(233,412)(234,411)(235,410)(236,409)(237,408)(238,407)(239,406)(240,405)(241,404)(242,403)(243,402)(244,401)(245,400)(246,399)(247,398)(248,397)(249,396)(250,395)(251,394)(252,393)(253,392)(254,391)(255,390)(256,389)(257,388)(258,387)(259,386)(260,385)(261,384)(262,383)(263,382)(264,381)(265,380)(266,379)(267,378)(268,377)(269,376)(270,375)(271,374)(272,373)(273,372)(274,371)(275,370)(276,369)(277,368)(278,367)(279,366)(280,365)(281,364)(282,363)(283,362)(284,361)(285,360)(286,359)(287,358)(288,357)(289,356)(290,355)(291,354)(292,353)(293,352)(294,351)(295,350)(296,349)(297,348)(298,347)(299,346)(300,345)(301,344)(302,343)(303,342)(304,341)(305,340)(306,339)(307,338)(308,337)(309,336)(310,335)(311,334)(312,333)(313,332)(314,331)(315,330)(316,329)(317,328)(318,327)(319,326)(320,325)(321,324)(322,323)

Polycyclic presentation:

2,11,613,2|a^2=d,b^a=b^10,c^a=c^612

Conjugacy classes & character table

6746 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₄₈₆
maximal normal subgroup of	Dic₁₃₄₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26972, 4);

Dic6743 #41004222