Dic₃₁₄₁ #27160239

order	12564
automorphism	Aut(G) order 13116816
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₁₄₁
unambiguous semidirect product	C₉⋊Dic₃₄₉

Splittings

derived, Hall	C₉×C₃₄₉ ⋊ C₄
Sylow	C₃₄₉ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₃₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₁₄₁
commutator, hypocenter	C₉×C₃₄₉	C₄
Frattini	C₂×C₃	D₁₀₄₇
Fitting, cosocle	C₂×C₉×C₃₄₉	C₂
socle, prime radical	C₂×C₃×C₃₄₉	D₃
LCS	G ▷ C₉×C₃₄₉
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 362
transitive degree	μ_t(G) = 12564

A minimal faithful permutation representation — 2 generators on 362 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,362)(16,361)(17,360)(18,359)(19,358)(20,357)(21,356)(22,355)(23,354)(24,353)(25,352)(26,351)(27,350)(28,349)(29,348)(30,347)(31,346)(32,345)(33,344)(34,343)(35,342)(36,341)(37,340)(38,339)(39,338)(40,337)(41,336)(42,335)(43,334)(44,333)(45,332)(46,331)(47,330)(48,329)(49,328)(50,327)(51,326)(52,325)(53,324)(54,323)(55,322)(56,321)(57,320)(58,319)(59,318)(60,317)(61,316)(62,315)(63,314)(64,313)(65,312)(66,311)(67,310)(68,309)(69,308)(70,307)(71,306)(72,305)(73,304)(74,303)(75,302)(76,301)(77,300)(78,299)(79,298)(80,297)(81,296)(82,295)(83,294)(84,293)(85,292)(86,291)(87,290)(88,289)(89,288)(90,287)(91,286)(92,285)(93,284)(94,283)(95,282)(96,281)(97,280)(98,279)(99,278)(100,277)(101,276)(102,275)(103,274)(104,273)(105,272)(106,271)(107,270)(108,269)(109,268)(110,267)(111,266)(112,265)(113,264)(114,263)(115,262)(116,261)(117,260)(118,259)(119,258)(120,257)(121,256)(122,255)(123,254)(124,253)(125,252)(126,251)(127,250)(128,249)(129,248)(130,247)(131,246)(132,245)(133,244)(134,243)(135,242)(136,241)(137,240)(138,239)(139,238)(140,237)(141,236)(142,235)(143,234)(144,233)(145,232)(146,231)(147,230)(148,229)(149,228)(150,227)(151,226)(152,225)(153,224)(154,223)(155,222)(156,221)(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)(164,213)(165,212)(166,211)(167,210)(168,209)(169,208)(170,207)(171,206)(172,205)(173,204)(174,203)(175,202)(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)(182,195)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)

Polycyclic presentation:

2,3,349,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^348,e^a=e^2

Conjugacy classes & character table

3144 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₂₈₂, Dic₉₄₂₃, Dic₁₅₇₀₅
maximal normal subgroup of	Dic₆₂₈₂
maximal quotient of	Dic₉₄₂₃, Dic₁₅₇₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12564, 6);

Dic3141 #27160239