#72253472

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#9497 ⋊ C₂
Sylow	C₃ ⋊ #245229

Subgroups

	subgroup	G/H
center	C₂²	#74884
commutator, cosocle	C₂⁵×C₃	C₂⁴
Frattini	C₂⁵	C₂³×D₃
Fitting	C₃×#9497	C₂
socle	C₂²×C₃	#2403
hypercenter	#9497	D₃
hypocenter	C₃	#245229
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #9497

6152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^9,3|b^2=i,c^2=e,d^2=h,b^a=bg,c^a=cf,c^b=ci,d^a=dh,d^b=di,d^c=deh,e^a=eh,f^b=fi,f^c=fh,f^d=fh,g^c=gi,j^d=j^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#245229
maximal normal subgroup of	D₃×#245229
maximal quotient of	#72270089

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