#72253469

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#14357 ⋊ C₂
Sylow	C₃ ⋊ #581603

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁴×D₃
commutator	C₂³×C₃	C₂⁴×C₄
Fitting	C₃×#14357	C₂
socle, cosocle	C₂⁴×C₃	C₂⁵
prime radical	C₂²×#5592	C₂
hypercenter	#14357	D₃
hypocenter	C₃	#581603
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ #14357

12292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|a^2=f,b^2=fgi,b^a=bg,c^a=cgh,c^b=cg,d^a=dgi,d^b=dg,e^a=ei,e^b=ehi,e^c=ei,j^c=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#581603, #72270557, #72271005
maximal normal subgroup of	D₃×#581603, #72270557, #72271005

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