Dic₃₃₅₃ #27211794

order	13412
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₃₅₃
unambiguous semidirect product	C₇⋊Dic₄₇₉
unambiguous semidirect product	C₄₇₉⋊Dic₇

Splittings

derived, Hall	C₇×C₄₇₉ ⋊ C₄
Sylow	C₄₇₉ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₄₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₃₅₃
commutator, hypocenter	C₇×C₄₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₄₇₉	C₂
LCS	G ▷ C₇×C₄₇₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 490
transitive degree	μ_t(G) = 13412

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

Polycyclic presentation:

2,7,479,2|a^2=d,b^a=b^6,c^a=c^478

Conjugacy classes & character table

3356 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₇₀₆
maximal normal subgroup of	Dic₆₇₀₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13412, 4);

Dic3353 #27211794