Dic₄₂₉₁ #31769786

order	17164
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₂₉₁
unambiguous semidirect product	C₇⋊Dic₆₁₃

Splittings

derived, Hall	C₇×C₆₁₃ ⋊ C₄
Sylow	C₆₁₃ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₆₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₂₉₁
commutator, hypocenter	C₇×C₆₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₆₁₃	C₂
LCS	G ▷ C₇×C₆₁₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 624
transitive degree	μ_t(G) = 17164

A minimal faithful permutation representation — 2 generators on 624 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,624)(14,623)(15,622)(16,621)(17,620)(18,619)(19,618)(20,617)(21,616)(22,615)(23,614)(24,613)(25,612)(26,611)(27,610)(28,609)(29,608)(30,607)(31,606)(32,605)(33,604)(34,603)(35,602)(36,601)(37,600)(38,599)(39,598)(40,597)(41,596)(42,595)(43,594)(44,593)(45,592)(46,591)(47,590)(48,589)(49,588)(50,587)(51,586)(52,585)(53,584)(54,583)(55,582)(56,581)(57,580)(58,579)(59,578)(60,577)(61,576)(62,575)(63,574)(64,573)(65,572)(66,571)(67,570)(68,569)(69,568)(70,567)(71,566)(72,565)(73,564)(74,563)(75,562)(76,561)(77,560)(78,559)(79,558)(80,557)(81,556)(82,555)(83,554)(84,553)(85,552)(86,551)(87,550)(88,549)(89,548)(90,547)(91,546)(92,545)(93,544)(94,543)(95,542)(96,541)(97,540)(98,539)(99,538)(100,537)(101,536)(102,535)(103,534)(104,533)(105,532)(106,531)(107,530)(108,529)(109,528)(110,527)(111,526)(112,525)(113,524)(114,523)(115,522)(116,521)(117,520)(118,519)(119,518)(120,517)(121,516)(122,515)(123,514)(124,513)(125,512)(126,511)(127,510)(128,509)(129,508)(130,507)(131,506)(132,505)(133,504)(134,503)(135,502)(136,501)(137,500)(138,499)(139,498)(140,497)(141,496)(142,495)(143,494)(144,493)(145,492)(146,491)(147,490)(148,489)(149,488)(150,487)(151,486)(152,485)(153,484)(154,483)(155,482)(156,481)(157,480)(158,479)(159,478)(160,477)(161,476)(162,475)(163,474)(164,473)(165,472)(166,471)(167,470)(168,469)(169,468)(170,467)(171,466)(172,465)(173,464)(174,463)(175,462)(176,461)(177,460)(178,459)(179,458)(180,457)(181,456)(182,455)(183,454)(184,453)(185,452)(186,451)(187,450)(188,449)(189,448)(190,447)(191,446)(192,445)(193,444)(194,443)(195,442)(196,441)(197,440)(198,439)(199,438)(200,437)(201,436)(202,435)(203,434)(204,433)(205,432)(206,431)(207,430)(208,429)(209,428)(210,427)(211,426)(212,425)(213,424)(214,423)(215,422)(216,421)(217,420)(218,419)(219,418)(220,417)(221,416)(222,415)(223,414)(224,413)(225,412)(226,411)(227,410)(228,409)(229,408)(230,407)(231,406)(232,405)(233,404)(234,403)(235,402)(236,401)(237,400)(238,399)(239,398)(240,397)(241,396)(242,395)(243,394)(244,393)(245,392)(246,391)(247,390)(248,389)(249,388)(250,387)(251,386)(252,385)(253,384)(254,383)(255,382)(256,381)(257,380)(258,379)(259,378)(260,377)(261,376)(262,375)(263,374)(264,373)(265,372)(266,371)(267,370)(268,369)(269,368)(270,367)(271,366)(272,365)(273,364)(274,363)(275,362)(276,361)(277,360)(278,359)(279,358)(280,357)(281,356)(282,355)(283,354)(284,353)(285,352)(286,351)(287,350)(288,349)(289,348)(290,347)(291,346)(292,345)(293,344)(294,343)(295,342)(296,341)(297,340)(298,339)(299,338)(300,337)(301,336)(302,335)(303,334)(304,333)(305,332)(306,331)(307,330)(308,329)(309,328)(310,327)(311,326)(312,325)(313,324)(314,323)(315,322)(316,321)(317,320)(318,319)

Polycyclic presentation:

2,7,613,2|a^2=d,b^a=b^6,c^a=c^612

Conjugacy classes & character table

4294 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₅₈₂, Dic₁₂₈₇₃
maximal normal subgroup of	Dic₈₅₈₂
maximal quotient of	Dic₁₂₈₇₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17164, 4);

Dic4291 #31769786