#72248686

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#8101 ⋊ C₂
Sylow	C₃ ⋊ #201664

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4665
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂⁵	C₂³×D₃
Fitting	C₃×#8101	C₂
socle	C₂²×C₃	C₂×#386
prime radical	C₂×#73852	C₂
cosocle	C₂⁵×C₃	C₂⁴
hypercenter	#8101	D₃
hypocenter	C₃	#201664
LCS	G ▷ C₂⁴×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ #8101

7936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|b^2=ef,c^2=i,b^a=be,c^a=cg,c^b=ci,d^b=dhi,e^b=eh,e^c=ei,f^a=fh,f^b=fh,f^c=fi,g^b=gi,j^a=j^2,j^b=j^2,j^c=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248696, D₃×#201664, #72253433
maximal normal subgroup of	#72248696, D₃×#201664, #72253433
maximal quotient of	#72255088

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