#72228849

order	6561
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁶ · Schur multiplier (H₂) = C₃¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁶
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ C₁

80164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 162
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 6 generators on 162 points:

(1,2,6)(3,7,18)(4,8,19)(5,10,20)(9,28,17)(11,21,38)(12,22,39)(13,24,49)(14,23,40)(15,25,41)(16,27,42)(26,52,37)(29,43,58)(30,45,60)(31,46,61)(32,48,70)(33,47,62)(34,44,59)(35,50,64)(36,51,63)(53,65,73)(54,67,75)(55,69,77)(56,68,76)(57,66,74)(71,79,81)(72,78,80)(82,83,87)(84,88,98)(85,89,99)(86,90,101)(91,102,118)(92,103,119)(93,104,120)(94,105,122)(95,107,124)(96,106,123)(97,108,100)(109,125,139)(110,126,140)(111,127,141)(112,128,142)(113,129,143)(114,131,121)(115,133,146)(116,132,145)(117,130,144)(134,147,154)(135,148,155)(136,149,156)(137,150,157)(138,151,158)(152,159,161)(153,160,162);(1,3,11)(2,7,21)(4,12,29)(5,14,30)(6,18,38)(8,22,43)(9,24,44)(10,23,45)(13,34,17)(15,31,53)(16,33,54)(19,39,58)(20,40,60)(25,46,65)(26,48,66)(27,47,67)(28,49,59)(32,57,37)(35,55,72)(36,56,71)(41,61,73)(42,62,75)(50,69,78)(51,68,79)(52,70,74)(63,76,81)(64,77,80)(82,84,91)(83,88,102)(85,92,109)(86,93,111)(87,98,118)(89,103,125)(90,104,127)(94,112,134)(95,113,136)(96,117,135)(97,114,110)(99,119,139)(100,121,140)(101,120,141)(105,128,147)(106,130,148)(107,129,149)(108,131,126)(115,138,152)(116,137,153)(122,142,154)(123,144,155)(124,143,156)(132,150,160)(133,151,159)(145,157,162)(146,158,161);(1,4,15)(2,8,25)(3,12,31)(5,16,36)(6,19,41)(7,22,46)(9,26,50)(10,27,51)(11,29,53)(13,32,55)(14,33,56)(17,37,35)(18,39,61)(20,42,63)(21,43,65)(23,47,68)(24,48,69)(28,52,64)(30,54,71)(34,57,72)(38,58,73)(40,62,76)(44,66,78)(45,67,79)(49,70,77)(59,74,80)(60,75,81)(82,85,94)(83,89,105)(84,92,112)(86,95,116)(87,99,122)(88,103,128)(90,107,132)(91,109,134)(93,113,137)(96,115,97)(98,119,142)(100,123,146)(101,124,145)(102,125,147)(104,129,150)(106,133,108)(110,135,152)(111,136,153)(114,117,138)(118,139,154)(120,143,157)(121,144,158)(126,148,159)(127,149,160)(130,151,131)(140,155,161)(141,156,162);(2,9,10)(3,13,14)(6,20,28)(7,23,24)(8,26,27)(11,30,34)(12,32,33)(19,42,52)(22,47,48)(25,50,51)(29,54,57)(31,55,56)(38,59,60)(41,63,64)(46,68,69)(53,71,72)(58,74,75)(73,80,81)(84,93,114)(88,104,131)(91,110,111)(92,113,117)(98,120,121)(102,126,127)(103,129,130)(109,135,136)(112,137,138)(118,140,141)(119,143,144)(125,148,149)(128,150,151)(134,152,153)(139,155,156)(142,157,158)(147,159,160)(154,161,162);(2,9,10)(6,20,28)(7,24,23)(8,26,27)(18,40,49)(19,42,52)(21,44,45)(22,48,47)(25,50,51)(38,60,59)(39,62,70)(41,63,64)(43,66,67)(46,69,68)(58,75,74)(61,76,77)(65,78,79)(73,81,80)(83,90,108)(85,95,96)(87,100,101)(88,104,131)(89,106,107)(92,113,117)(94,115,116)(98,121,120)(102,127,126)(103,130,129)(109,136,135)(112,138,137)(118,140,141)(122,145,146)(125,148,149)(134,152,153)(142,157,158)(154,162,161);(4,16,37)(8,27,26)(12,33,32)(15,35,36)(19,42,52)(22,47,48)(25,50,51)(29,54,57)(31,55,56)(39,62,70)(41,64,63)(43,67,66)(46,69,68)(53,72,71)(58,75,74)(61,77,76)(65,78,79)(73,80,81)(84,93,114)(85,96,95)(88,104,131)(89,106,107)(91,110,111)(94,116,115)(98,120,121)(99,123,124)(102,126,127)(105,132,133)(109,136,135)(112,138,137)(118,140,141)(122,145,146)(125,149,148)(128,151,150)(139,156,155)(142,158,157)

Polycyclic presentation:

3^8|d^a=dg,d^b=dgh^2,e^a=egh^2,e^c=eh^2,f^b=fh^2,f^c=fg^2h

Conjugacy classes & character table

737 conjugacy classes — view the full character table.

Relationships

maximal quotient of

H(3,𝔽₂₇)

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", "h");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);