D₁₆₁₈₄ #72272540

order	32368
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₁₈₄

Splittings

Fitting	C₇×C₈×C₂₈₉ ⋊ C₂
Hall	C₇×C₂₈₉ ⋊ D₈
Sylow	C₂₈₉ ⋊ D₅₆
Sylow	C₇ ⋊ D₂₃₁₂

Subgroups

	subgroup	G/H
center	C₂	D₈₀₉₂
commutator, cosocle	C₄×C₇×C₂₈₉	C₂²
Frattini	C₄×C₁₇	C₂×D₁₁₉
Fitting	C₇×C₈×C₂₈₉	C₂
socle	C₂×C₇×C₁₇	D₆₈
hypercenter	C₈	D₂₀₂₃
hypocenter	C₇×C₂₈₉	D₈
LCS	G ▷ C₄×C₇×C₂₈₉ ▷ C₂×C₇×C₂₈₉ ▷ C₇×C₂₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 304
transitive degree	μ_t(G) = 16184

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

Polycyclic presentation:

2^4,7,17^2|b^2=cd,c^2=d,f^17=g,b^a=bc,c^a=cd,e^a=e^6,f^a=f^16g^16,g^a=g^16

Conjugacy classes & character table

8095 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₂₃₆₈
maximal normal subgroup of	D₃₂₃₆₈
maximal quotient of	Dic₁₆₁₈₄, D₃₂₃₆₈

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, 7, 17, 17]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^16*F.7^16);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D16184 #72272540