#72252338

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#1781 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#12705

Subgroups

	subgroup	G/H
center	C₂	#72252263
commutator	C₂⁴×C₃²	C₂³×C₄
Frattini	C₂²×D₄	C₂²×D₃²
Fitting	C₃²×#1781	C₂²
socle	C₂×C₃²	C₂×#1781
prime radical	#72215172	C₂
cosocle	C₂²×C₃²×D₄	C₂⁴
hypercenter	#1781	D₃²
hypocenter	C₃²	C₂×#12705
LCS	G ▷ C₂⁴×C₃² ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ #1781

19887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^9,3^2|a^2=h,c^2=fgi,g^2=i,c^a=ce,c^b=cef,e^a=ei,e^b=ei,e^c=eh,f^b=fi,f^c=fh,g^a=gh,g^b=gi,g^c=gh,g^e=gi,g^f=gi,h^c=hi,j^a=j^2,j^d=j^2,k^a=j^2k,k^d=k^2

Conjugacy classes & character table

168 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);;
gap> SetPower(coll, 3, F.6*F.7*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*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);