#72253518

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #359509

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5220
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting, prime radical	C₂²×C₃×#535	C₂
socle	C₂²×C₃	C₂×#386
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	C₂²×#535	D₃
hypocenter	C₃	#359509
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#535

9420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|a^2=h,b^2=efi,e^2=i,f^2=i,b^a=be,c^b=cg,d^a=dh,d^b=dg,e^d=ei,f^a=fi,f^d=fi,h^b=hi,j^b=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253520, D₃×#359509, #72253559, …
maximal normal subgroup of	#72253520, D₃×#359509, #72253559, …

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