#72265524

order	19683
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₃¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₃²	#14552185
commutator, Frattini, cosocle	C₃⁵	C₃⁴
LCS	G ▷ C₃⁵ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁵ ▷ C₃² ▷ C₁

41706 conjugacy classes of subgroups; only the 433 normal subgroups are shown.

normal subgroups only

This group has 41706 conjugacy classes of subgroups — too many to draw. Only its 433 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) = 54
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 4 generators on 54 points:

(1,2,5)(3,6,12)(4,7,14)(8,16,22)(9,17,21)(10,15,23)(11,18,13)(19,24,27)(20,25,26);(6,15,17)(12,21,23)(16,25,24)(22,27,26)(28,29,32)(30,33,39)(31,34,41)(35,43,49)(36,44,48)(37,42,50)(38,45,40)(46,51,54)(47,52,53);(1,3,8)(2,6,16)(4,10,19)(5,12,22)(7,15,24)(9,20,11)(13,21,26)(14,23,27)(17,25,18)(33,42,44)(39,48,50)(43,52,51)(49,54,53);(28,30,35)(29,33,43)(31,37,46)(32,39,49)(34,42,51)(36,47,38)(40,48,53)(41,50,54)(44,52,45)

Polycyclic presentation:

3^9|b^a=be,c^b=cf,d^c=dg,e^c=eh,f^a=fh^2,f^d=fi,g^b=gi^2

Conjugacy classes & character table

339 conjugacy classes — view the full character table.

Relationships

maximal quotient of

UT(5,𝔽₃)

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", "i");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);