#72248103

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #115690

Subgroups

	subgroup	G/H
center	C₂²	#72131
commutator	C₂⁵×C₃	C₂²×C₄
Frattini	C₂⁶	C₂²×D₃
Fitting, prime radical	C₂×C₃×#2846	C₂
socle	C₂²×C₃	#1438
cosocle	C₂⁶×C₃	C₂³
hypercenter	C₂×#2846	D₃
hypocenter	C₃	#115690
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ C₂×#2846

18330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^9,3|a^2=f,b^2=d,b^a=bg,c^a=cfh,c^b=ce,d^a=dh,e^a=ehi,f^b=fhi,g^a=ghi,g^b=gh,j^b=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#115690, #72249061, #72253431
maximal normal subgroup of	D₃×#115690, #72249061, #72253431
maximal quotient of	#72255190

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