#72265412

order	6561
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₃	#14550628
commutator, Frattini, cosocle	C₃²×H(3,𝔽₃)	C₃³
LCS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃²×H(3,𝔽₃) ◁ G
LECS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃² ▷ C₃ ▷ C₁

3339 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 3339 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,5)(3,7,12)(4,8,13)(6,15,10)(9,18,24)(11,19,14)(16,26,22)(17,25,21)(20,27,23);(2,6,7)(5,12,15)(8,17,18)(13,24,25)(14,23,26)(16,27,19);(3,9,20)(10,22,21)(12,23,24)(15,25,26)

Polycyclic presentation:

3^8|b^a=bd,c^a=cf,c^b=ce,d^c=dg,e^a=eg,f^b=fg^2h,f^d=fh^2,g^a=gh

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72265419

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