#72271359

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×#535 ⋊ C₂
Sylow	C₃ ⋊ #687593

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁴×D₃
commutator, socle, cosocle	C₂⁴×C₃	C₂⁵
Fitting	C₂²×C₃×#535	C₂
hypercenter	C₂²×#535	D₃
hypocenter	C₃	#687593
LCS	G ▷ C₂⁴×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#535

17988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|d^2=f,c^a=ci,c^b=cf,d^a=di,e^a=eh,e^b=eg,e^d=ef,j^a=j^2

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#687593, #72271442, #72271583
maximal normal subgroup of	D₃×#687593, #72271442, #72271583

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