#72217355

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72216433 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂	#72217282
commutator, cosocle	#6499867	C₂²×C₃
Frattini	C₂⁷	C₂²×A₄
Fitting	#72216433	C₃
hypercenter	C₂²	#72215851
hypocenter	#62817	C₃×D₄
LCS	G ▷ #6499867 ▷ #62817
UCS	C₁ ◁ C₂ ◁ C₂²

At least 6595 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has at least 6595 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^9|e^2=l,f^2=l,b^a=bd,e^a=eh,e^b=ej,e^c=f,e^d=ej,f^a=fg,f^b=fi,f^c=efl,f^d=fi,f^e=fl,g^b=gi,g^c=ghk,g^e=gk,h^b=hj,h^c=g,h^f=hk,i^c=ijl,i^e=il,j^c=i,j^f=jl,k^b=kl

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72218792
maximal quotient of	#72252407

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);