#72269677

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#72224648 ⋊ C₂
Sylow	C₃ ⋊ #72224662

Subgroups

	subgroup	G/H
center	C₂	#72245089
commutator	C₃×C₃₂	C₂²×C₁₆
Frattini	#7330	C₂²×D₃
Fitting	C₃×#72224648	C₂
socle	C₂×C₃	#72224639
prime radical	#10716247	C₈
cosocle	C₃×#7330	C₂³
hypercenter	#72224648	D₃
hypocenter	C₃	#72224662
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1311 ◁ #7330 ◁ #72224648

436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Polycyclic presentation:

2^11,3|a^2=e,b^2=d,c^2=df,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=begijk,c^a=cgijk,d^a=dijk,e^b=eg,e^c=ei,e^d=ej,f^a=fjk,f^e=fk,g^b=gi,g^c=gj,g^d=gk,h^a=hk,i^b=ij,i^c=ik,j^b=jk,l^c=l^2

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72224662
maximal normal subgroup of	D₃×#72224662

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