#72265023

order	6561
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₃¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₃	#14552185
commutator, Frattini, cosocle	C₃⁴	C₃⁴
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

8184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 4 generators on 81 points:

(1,2,6)(3,7,17)(4,8,18)(5,9,20)(10,21,40)(11,22,41)(12,23,42)(13,24,43)(14,25,44)(15,27,45)(16,28,19)(26,46,39)(29,47,61)(30,48,62)(31,49,64)(32,50,65)(33,52,66)(34,53,63)(35,51,67)(36,54,68)(37,56,69)(38,55,70)(57,71,76)(58,72,77)(59,73,78)(60,74,79)(75,80,81);(1,3,10)(2,7,21)(4,11,31)(5,13,30)(6,17,40)(8,22,49)(9,24,48)(12,34,16)(14,36,57)(15,33,29)(18,41,64)(19,42,63)(20,43,62)(23,53,28)(25,54,71)(26,51,72)(27,52,47)(32,59,37)(35,58,39)(38,60,75)(44,68,76)(45,66,61)(46,67,77)(50,73,56)(55,74,80)(65,78,69)(70,79,81);(1,4,14)(2,8,25)(3,11,32)(5,15,38)(6,18,44)(7,22,50)(9,27,55)(10,29,57)(12,35,60)(13,33,36)(16,39,37)(17,41,65)(19,46,69)(20,45,70)(21,47,71)(23,51,74)(24,52,54)(26,56,28)(30,58,75)(31,59,34)(40,61,76)(42,67,79)(43,66,68)(48,72,80)(49,73,53)(62,77,81)(63,64,78);(8,26,27)(18,45,46)(22,51,52)(25,55,56)(41,66,67)(44,69,70)(47,49,72)(50,54,74)(61,77,64)(65,79,68)(71,80,73)(76,78,81)

Polycyclic presentation:

3^8|c^b=ce,d^a=df,d^c=dg,e^b=eh,e^c=eh^2,f^c=fh^2,g^a=gh^2

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72277684, #72277778
maximal subgroup of	#72265118, #72277684, #72277778
maximal normal subgroup of	#72265118, #72277684, #72277778

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