#72255287

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13967 ⋊ C₂
Sylow	C₃ ⋊ #850465

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂⁴×D₃
commutator, socle, cosocle	C₂⁴×C₃	C₂⁵
Fitting	C₃×#13967	C₂
hypercenter	#13967	D₃
hypocenter	C₃	#850465
LCS	G ▷ C₂⁴×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ #13967

12628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 384

Polycyclic presentation:

2^9,3|b^2=h,e^2=fghi,b^a=bf,c^a=ci,d^a=dg,e^a=ei,e^b=ef,e^c=efghi,e^d=egh,j^d=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255403, D₃×#850465, #72270816
maximal normal subgroup of	#72255403, D₃×#850465, #72270816

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