#72269876

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#23848 ⋊ C₂
Sylow	C₃ ⋊ #6453969

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃×D₄
commutator, cosocle	C₂²×C₃×C₄	C₂⁵
Frattini	C₂²×C₄	C₂⁴×D₃
Fitting	C₃×#23848	C₂
socle	C₂³×C₃	C₂³×D₄
hypercenter	#23848	D₃
hypocenter	C₃	#6453969
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ #23848

9176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|b^2=f,c^2=h,e^2=h,f^2=i,c^a=ch,c^b=cg,d^b=df,e^b=ei,f^d=fi,j^e=j^2

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#6453969
maximal normal subgroup of	D₃×#6453969
maximal quotient of	#72270183

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