D₁₃₅₈₄ #72235518

order	27168
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₈₄

Splittings

Fitting	C₃×C₁₆×C₂₈₃ ⋊ C₂
Hall	C₃×C₂₈₃ ⋊ D₁₆
Sylow	C₂₈₃ ⋊ D₄₈
Sylow	C₃ ⋊ D₄₅₂₈

Subgroups

	subgroup	G/H
center	C₂	D₆₇₉₂
commutator, cosocle	C₃×C₈×C₂₈₃	C₂²
Frattini	C₈	C₂×D₈₄₉
Fitting	C₃×C₁₆×C₂₈₃	C₂
socle	C₂×C₃×C₂₈₃	D₈
hypercenter	C₁₆	D₈₄₉
hypocenter	C₃×C₂₈₃	D₁₆
LCS	G ▷ C₃×C₈×C₂₈₃ ▷ C₃×C₄×C₂₈₃ ▷ C₂×C₃×C₂₈₃ ▷ C₃×C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 302
transitive degree	μ_t(G) = 13584

A minimal faithful permutation representation — 2 generators on 302 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,302)(22,301)(23,300)(24,299)(25,298)(26,297)(27,296)(28,295)(29,294)(30,293)(31,292)(32,291)(33,290)(34,289)(35,288)(36,287)(37,286)(38,285)(39,284)(40,283)(41,282)(42,281)(43,280)(44,279)(45,278)(46,277)(47,276)(48,275)(49,274)(50,273)(51,272)(52,271)(53,270)(54,269)(55,268)(56,267)(57,266)(58,265)(59,264)(60,263)(61,262)(62,261)(63,260)(64,259)(65,258)(66,257)(67,256)(68,255)(69,254)(70,253)(71,252)(72,251)(73,250)(74,249)(75,248)(76,247)(77,246)(78,245)(79,244)(80,243)(81,242)(82,241)(83,240)(84,239)(85,238)(86,237)(87,236)(88,235)(89,234)(90,233)(91,232)(92,231)(93,230)(94,229)(95,228)(96,227)(97,226)(98,225)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211)(113,210)(114,209)(115,208)(116,207)(117,206)(118,205)(119,204)(120,203)(121,202)(122,201)(123,200)(124,199)(125,198)(126,197)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,182)(142,181)(143,180)(144,179)(145,178)(146,177)(147,176)(148,175)(149,174)(150,173)(151,172)(152,171)(153,170)(154,169)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)

Polycyclic presentation:

2^5,3,283|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^282

Conjugacy classes & character table

6795 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₁₆₈
maximal normal subgroup of	D₂₇₁₆₈
maximal quotient of	Dic₁₃₅₈₄, D₂₇₁₆₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 283]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^282);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13584 #72235518