Dic₄₀₃₉ #30633394

order	16156
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₃₉
unambiguous semidirect product	C₇⋊Dic₅₇₇

Splittings

derived, Hall	C₇×C₅₇₇ ⋊ C₄
Sylow	C₅₇₇ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₅₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₃₉
commutator, hypocenter	C₇×C₅₇₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₅₇₇	C₂
LCS	G ▷ C₇×C₅₇₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 588
transitive degree	μ_t(G) = 16156

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

Polycyclic presentation:

2,7,577,2|a^2=d,b^a=b^6,c^a=c^576

Conjugacy classes & character table

4042 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₇₈
maximal normal subgroup of	Dic₈₀₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16156, 4);

Dic4039 #30633394