D₁₈₅₀₄ #72273591

order	37008
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₅₀₄

Splittings

Fitting	C₈×C₉×C₂₅₇ ⋊ C₂
Hall	C₉×C₂₅₇ ⋊ D₈
Sylow	C₂₅₇ ⋊ D₇₂
Sylow	C₉ ⋊ D₂₀₅₆

Subgroups

	subgroup	G/H
center	C₂	D₉₂₅₂
commutator, cosocle	C₄×C₉×C₂₅₇	C₂²
Frattini	C₃×C₄	C₂×D₇₇₁
Fitting	C₈×C₉×C₂₅₇	C₂
socle	C₂×C₃×C₂₅₇	D₁₂
hypercenter	C₈	D₂₃₁₃
hypocenter	C₉×C₂₅₇	D₈
LCS	G ▷ C₄×C₉×C₂₅₇ ▷ C₂×C₉×C₂₅₇ ▷ C₉×C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 274
transitive degree	μ_t(G) = 18504

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

Polycyclic presentation:

2^4,3,257,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^256,g^a=g^2

Conjugacy classes & character table

9255 conjugacy classes. The character table is too large to display.

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, 3, 257, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, 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.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^256);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18504 #72273591