#72219952

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×D₄×3₋¹⁺² ⋊ C₂²
Sylow	C₃×3₋¹⁺² ⋊ H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₃×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₃×C₉	C₂⁴×C₃
Frattini	C₂×C₃	C₂²×C₃×D₃²
Fitting	C₃×D₄×3₋¹⁺²	C₂²
socle	C₂×C₃²	C₂³×C₃×D₃
hypercenter	D₄	D₃×AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ D₄

1241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2,3^3|a^2=f,g^3=i,c^a=cf,d^b=df,g^b=g^2i^2,g^d=g^2i^2,g^e=gi,h^b=hi^2,h^d=h^2,i^b=i^2,i^d=i^2

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219955, #72219989
maximal normal subgroup of	#72219989
maximal quotient of	#72219955

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