Dic₂₈₁₇ #26014663

order	11268
automorphism	Aut(G) order 10546848
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₈₁₇
unambiguous semidirect product	C₉⋊Dic₃₁₃

Splittings

derived, Hall	C₉×C₃₁₃ ⋊ C₄
Sylow	C₃₁₃ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₃₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₈₁₇
commutator, hypocenter	C₉×C₃₁₃	C₄
Frattini	C₂×C₃	D₉₃₉
Fitting, cosocle	C₂×C₉×C₃₁₃	C₂
socle, prime radical	C₂×C₃×C₃₁₃	D₃
LCS	G ▷ C₉×C₃₁₃
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 326
transitive degree	μ_t(G) = 11268

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

Polycyclic presentation:

2,3,313,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^312,e^a=e^2

Conjugacy classes & character table

2820 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₆₃₄, Dic₈₄₅₁, Dic₁₄₀₈₅
maximal normal subgroup of	Dic₅₆₃₄
maximal quotient of	Dic₈₄₅₁, Dic₁₄₀₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11268, 6);

Dic2817 #26014663