Dic₃₇₀₇ #28415282

order	14828
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₇₀₇
unambiguous semidirect product	C₁₁⋊Dic₃₃₇

Splittings

derived, Hall	C₁₁×C₃₃₇ ⋊ C₄
Sylow	C₃₃₇ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₃₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₇₀₇
commutator, hypocenter	C₁₁×C₃₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₃₃₇	C₂
LCS	G ▷ C₁₁×C₃₃₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 352
transitive degree	μ_t(G) = 14828

A minimal faithful permutation representation — 2 generators on 352 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,352)(18,351)(19,350)(20,349)(21,348)(22,347)(23,346)(24,345)(25,344)(26,343)(27,342)(28,341)(29,340)(30,339)(31,338)(32,337)(33,336)(34,335)(35,334)(36,333)(37,332)(38,331)(39,330)(40,329)(41,328)(42,327)(43,326)(44,325)(45,324)(46,323)(47,322)(48,321)(49,320)(50,319)(51,318)(52,317)(53,316)(54,315)(55,314)(56,313)(57,312)(58,311)(59,310)(60,309)(61,308)(62,307)(63,306)(64,305)(65,304)(66,303)(67,302)(68,301)(69,300)(70,299)(71,298)(72,297)(73,296)(74,295)(75,294)(76,293)(77,292)(78,291)(79,290)(80,289)(81,288)(82,287)(83,286)(84,285)(85,284)(86,283)(87,282)(88,281)(89,280)(90,279)(91,278)(92,277)(93,276)(94,275)(95,274)(96,273)(97,272)(98,271)(99,270)(100,269)(101,268)(102,267)(103,266)(104,265)(105,264)(106,263)(107,262)(108,261)(109,260)(110,259)(111,258)(112,257)(113,256)(114,255)(115,254)(116,253)(117,252)(118,251)(119,250)(120,249)(121,248)(122,247)(123,246)(124,245)(125,244)(126,243)(127,242)(128,241)(129,240)(130,239)(131,238)(132,237)(133,236)(134,235)(135,234)(136,233)(137,232)(138,231)(139,230)(140,229)(141,228)(142,227)(143,226)(144,225)(145,224)(146,223)(147,222)(148,221)(149,220)(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)(159,210)(160,209)(161,208)(162,207)(163,206)(164,205)(165,204)(166,203)(167,202)(168,201)(169,200)(170,199)(171,198)(172,197)(173,196)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)

Polycyclic presentation:

2,11,337,2|a^2=d,b^a=b^10,c^a=c^336

Conjugacy classes & character table

3710 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₄₁₄
maximal normal subgroup of	Dic₇₄₁₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14828, 4);

Dic3707 #28415282