#72254696

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄⋊C₄×3₋¹⁺² ⋊ C₂
Sylow	C₃×3₋¹⁺² ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₃⋊AGL(1,ℤ/9)
commutator	C₂×C₃×C₉	C₂²×C₃×C₄
Frattini	C₂²×C₃	C₂²×C₃×C₃⋊D₃
Fitting	C₃×C₄⋊C₄×3₋¹⁺²	C₂
socle	C₂²×C₃²	C₂²×C₃×D₃
prime radical	C₂×C₄⋊(C₃⋊AGL(1,ℤ/9))	C₂
cosocle	C₂²×C₃×C₉	C₂³×C₃
hypercenter	C₄⋊C₄	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃×C₉	C₃×C₄×D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

1057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 432

Polycyclic presentation:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254708, C₄×D₄×C₃⋊AGL(1,ℤ/9), #72254769
maximal normal subgroup of	#72254708, C₄×D₄×C₃⋊AGL(1,ℤ/9), #72254769
maximal quotient of	#72254708

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