#72243296

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14552588 ⋊ C₂²
derived	C₉³ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉³	C₂²×C₃
Frattini, socle	C₃³	C₃×D₃×C₃⋊D₃
Fitting	#14552588	C₂²
LCS	G ▷ C₉³
UCS	C₁

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 486

Polycyclic presentation:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#92314, #72243391
maximal normal subgroup of	AGL(1,ℤ/9)×#92314, #72243391

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