Dic₄₀₄₃ #30633598

order	16172
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₄₃
unambiguous semidirect product	C₃₁₁⋊Dic₁₃

Splittings

derived, Hall	C₁₃×C₃₁₁ ⋊ C₄
Sylow	C₃₁₁ ⋊ Dic₁₃
Sylow	C₁₃ ⋊ Dic₃₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₄₃
commutator, hypocenter	C₁₃×C₃₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₃₁₁	C₂
LCS	G ▷ C₁₃×C₃₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 328
transitive degree	μ_t(G) = 16172

A minimal faithful permutation representation — 2 generators on 328 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,328)(20,327)(21,326)(22,325)(23,324)(24,323)(25,322)(26,321)(27,320)(28,319)(29,318)(30,317)(31,316)(32,315)(33,314)(34,313)(35,312)(36,311)(37,310)(38,309)(39,308)(40,307)(41,306)(42,305)(43,304)(44,303)(45,302)(46,301)(47,300)(48,299)(49,298)(50,297)(51,296)(52,295)(53,294)(54,293)(55,292)(56,291)(57,290)(58,289)(59,288)(60,287)(61,286)(62,285)(63,284)(64,283)(65,282)(66,281)(67,280)(68,279)(69,278)(70,277)(71,276)(72,275)(73,274)(74,273)(75,272)(76,271)(77,270)(78,269)(79,268)(80,267)(81,266)(82,265)(83,264)(84,263)(85,262)(86,261)(87,260)(88,259)(89,258)(90,257)(91,256)(92,255)(93,254)(94,253)(95,252)(96,251)(97,250)(98,249)(99,248)(100,247)(101,246)(102,245)(103,244)(104,243)(105,242)(106,241)(107,240)(108,239)(109,238)(110,237)(111,236)(112,235)(113,234)(114,233)(115,232)(116,231)(117,230)(118,229)(119,228)(120,227)(121,226)(122,225)(123,224)(124,223)(125,222)(126,221)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,210)(138,209)(139,208)(140,207)(141,206)(142,205)(143,204)(144,203)(145,202)(146,201)(147,200)(148,199)(149,198)(150,197)(151,196)(152,195)(153,194)(154,193)(155,192)(156,191)(157,190)(158,189)(159,188)(160,187)(161,186)(162,185)(163,184)(164,183)(165,182)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)

Polycyclic presentation:

2,13,311,2|a^2=d,b^a=b^12,c^a=c^310

Conjugacy classes & character table

4046 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₈₆
maximal normal subgroup of	Dic₈₀₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16172, 4);

Dic4043 #30633598