#72252397

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1438 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#8453

Subgroups

	subgroup	G/H
center	C₂²	#11800963
commutator	C₂²×C₃²×C₄	C₂³×C₄
Frattini	C₂³×C₄	C₂²×D₃²
Fitting	C₃²×#1438	C₂²
socle	C₂²×C₃²	C₂×#379
prime radical	#72215172	C₂
cosocle	C₂³×C₃²×C₄	C₂⁴
hypercenter	#1438	D₃²
hypocenter	C₃²	C₂×#8453
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1438

22791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^9,3^2|a^2=hi,b^2=e,f^2=i,g^2=i,b^a=bf,c^b=cefgi,e^a=eh,f^a=fi,f^b=fhi,f^c=fi,g^a=gh,g^b=ghi,g^c=gi,j^a=j^2,j^d=j^2,k^a=j^2k,k^d=k^2

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.8*F.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 4, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);