Dic₃₁₉₉ #27165348

order	12796
automorphism	Aut(G) order 17504928
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₁₉₉
unambiguous semidirect product	C₇⋊Dic₄₅₇

Splittings

derived, Hall	C₇×C₄₅₇ ⋊ C₄
Sylow	C₄₅₇ ⋊ Dic₇
Sylow	C₇ ⋊ Dic₄₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₁₉₉
commutator, hypocenter	C₇×C₄₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₄₅₇	C₂
LCS	G ▷ C₇×C₄₅₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 468
transitive degree	μ_t(G) = 12796

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

Polycyclic presentation:

2,7,457,2|a^2=d,b^a=b^6,c^a=c^456

Conjugacy classes & character table

3202 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₃₁₉₉, Dic₉₅₉₇, #50307079, …
maximal normal subgroup of	C₃×Dic₃₁₉₉, #50307079, ΓL(1,𝔽₈)⋊Dic₄₅₇, …
maximal quotient of	C₃×Dic₃₁₉₉, Dic₉₅₉₇, Dic₁₅₉₉₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12796, 4);

Dic3199 #27165348