#72269902

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #72210566

Subgroups

	subgroup	G/H
center	C₂²	C₂×#72142
commutator	C₂²×C₃×D₄	C₂³×C₄
Frattini	C₂³×D₄	C₂³×D₃
Fitting, prime radical	C₂×C₃×#52004	C₂
socle	C₂²×C₃	C₂×#1790
cosocle	C₂³×C₃×D₄	C₂⁴
hypercenter	C₂×#52004	D₃
hypocenter	C₃	#72210566
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ C₂×#52004

16042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^10,3|a^2=ij,b^2=e,g^2=j,b^a=bg,c^b=cef,d^b=dh,e^a=ej,e^c=eh,f^a=fi,f^b=fh,f^c=fh,g^c=gi,g^f=gj,i^b=ij,k^b=k^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72210566, #72270192, #72271498
maximal normal subgroup of	D₃×#72210566, #72270192, #72271498

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);