#72338989

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237819
commutator, Frattini, cosocle	#20586	C₂⁴
prime radical	#72241389	C₂
LCS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628901 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

15650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Polycyclic presentation:

2^12|b^2=kl,c^2=eik,d^2=kl,f^2=i,g^2=l,b^a=bh,c^a=cf,c^b=ce,d^a=di,d^b=dkl,d^c=dg,e^a=eh,e^b=el,e^c=ek,e^d=ejkl,f^a=fi,f^b=fh,f^c=fhikl,f^d=fikl,f^e=fjk,g^a=gi,g^b=gjkl,g^c=gjk,g^e=gl,h^c=hj,h^d=hkl,h^f=hkl,h^g=hl,i^b=ikl,i^c=ikl,i^e=il,j^a=jk,j^f=jl,k^c=kl

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339011
maximal normal subgroup of	#72339011

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