Dic₂₈₅₃ #26037585

order	11412
automorphism	Aut(G) order 10818576
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₈₅₃
unambiguous semidirect product	C₉⋊Dic₃₁₇

Splittings

derived, Hall	C₉×C₃₁₇ ⋊ C₄
Sylow	C₃₁₇ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₃₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₈₅₃
commutator, hypocenter	C₉×C₃₁₇	C₄
Frattini	C₂×C₃	D₉₅₁
Fitting, cosocle	C₂×C₉×C₃₁₇	C₂
socle, prime radical	C₂×C₃×C₃₁₇	D₃
LCS	G ▷ C₉×C₃₁₇
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 330
transitive degree	μ_t(G) = 11412

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

Polycyclic presentation:

2,3,317,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^316,e^a=e^2

Conjugacy classes & character table

2856 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₇₀₆, Dic₈₅₅₉, Dic₁₄₂₆₅
maximal normal subgroup of	Dic₅₇₀₆
maximal quotient of	Dic₈₅₅₉, Dic₁₄₂₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11412, 4);

Dic2853 #26037585