#72270124

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2688 ⋊ C₂
Sylow	C₃ ⋊ #1420724

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁴×D₃
commutator, socle, cosocle	C₂⁴×C₃	C₂⁵
Fitting	C₂×C₃×#2688	C₂
hypercenter	C₂×#2688	D₃
hypocenter	C₃	#1420724
LCS	G ▷ C₂⁴×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#2688

12160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|a^2=h,c^b=cf,d^a=dh,d^b=di,d^c=dhi,e^a=ehi,e^c=eg,e^d=ei,j^b=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270150, D₃×#1420724, #72270952
maximal normal subgroup of	#72270150, D₃×#1420724, #72270952

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