#72243228

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃×C₉²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552588 ⋊ C₂
derived	C₉³ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉³	C₂×C₃
Frattini, socle	C₃³	C₃×C₃⋊(C₃⋊D₃)
Fitting	#14552588	C₂
LCS	G ▷ C₉³
UCS	C₁

1287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 729

Polycyclic presentation:

2,3^7|c^3=f,d^3=g,e^3=h,c^a=c^2f^2,c^b=cf^2,d^a=d^2g^2,d^b=dg^2,e^a=e^2h^2,e^b=eh,f^a=f^2,g^a=g^2,h^a=h^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243235, #72243296, #72243357
maximal normal subgroup of	#72243235, #72243296, #72243357

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