D₂₈₄₃₅ #72291997

order	56870
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₈₄₃₅

Splittings

Fitting, derived, Hall	C₅×C₄₇×C₁₂₁ ⋊ C₂
Hall	C₄₇×C₁₂₁ ⋊ D₅
Hall	C₅×C₁₂₁ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₁₂₁
Sylow	C₁₂₁ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₆₀₅
Sylow	C₅ ⋊ D₅₆₈₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₄₇×C₁₂₁	C₂
Frattini	C₁₁	D₂₅₈₅
socle	C₅×C₁₁×C₄₇	D₁₁
LCS	G ▷ C₅×C₄₇×C₁₂₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 173
transitive degree	μ_t(G) = 28435

A minimal faithful permutation representation — 2 generators on 173 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);(2,5)(3,4)(7,126)(8,125)(9,124)(10,123)(11,122)(12,121)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(128,173)(129,172)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)(139,162)(140,161)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)

Polycyclic presentation:

2,5,11,47,11|c^11=e,b^a=b^4,c^a=c^10e^10,d^a=d^46,e^a=e^10

Conjugacy classes & character table

14219 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");;
gap> coll := SingleCollector(F, [2, 5, 11, 47, 11]);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3^10*F.5^10);;
gap> SetConjugate(coll, 4, 1, F.4^46);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28435 #72291997