Dic₄₁₀₉ #30656163

order	16436
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₁₀₉
unambiguous semidirect product	C₇⋊Dic₅₈₇
unambiguous semidirect product	C₅₈₇⋊Dic₇

Splittings

derived, Hall	C₇×C₅₈₇ ⋊ C₄
Sylow	C₅₈₇ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₅₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₁₀₉
commutator, hypocenter	C₇×C₅₈₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₅₈₇	C₂
LCS	G ▷ C₇×C₅₈₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 598
transitive degree	μ_t(G) = 16436

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

Polycyclic presentation:

2,7,587,2|a^2=d,b^a=b^6,c^a=c^586

Conjugacy classes & character table

4112 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₂₁₈
maximal normal subgroup of	Dic₈₂₁₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16436, 4);

Dic4109 #30656163