#72243266

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	3₋¹⁺²×C₉⋊C₉ ⋊ C₂
derived	C₉² ⋊ C₂×C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#13091521
commutator, hypocenter	C₉²	C₂×C₃×C₉
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	3₋¹⁺²×C₉⋊C₉	C₂
prime radical	C₃×#92299	C₃
cosocle	C₃×C₉²	C₂×C₃²
LCS	G ▷ C₉²
UCS	C₁ ◁ C₃

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 162

Polycyclic presentation:

2,3^7|c^3=d,e^3=g,f^3=h,c^a=cg^2,e^a=e^2g^2,e^c=eg,f^a=f^2h^2,f^b=fh,g^a=g^2,h^a=h^2

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243278, AGL(1,ℤ/9)×D₉⋊C₉
maximal normal subgroup of	#72243278, AGL(1,ℤ/9)×D₉⋊C₉

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