#72340453

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#72210482
commutator	#2309	C₂³×C₄
Frattini, cosocle	#20586	C₂⁴
prime radical	#72340439	C₂
LCS	G ▷ #2309 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #72340431 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

At least 17538 conjugacy classes of subgroups; only the 393 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=hijk,d^2=f,e^2=ijl,b^a=bhijk,c^a=cl,c^b=cl,d^a=dg,d^b=de,d^c=dl,e^a=ei,e^b=eikl,e^d=eh,f^b=fhijk,f^e=fj,g^b=gijkl,g^e=gk,h^a=hl,h^b=hl,h^d=hj,h^e=hkl,h^f=hl,h^g=hl,i^b=ikl,i^d=ik,i^e=il,i^f=il,j^a=jl,j^b=jk,j^d=jl,j^e=jl,k^d=kl

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

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