#72287397

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72284488
commutator, Frattini, cosocle	C₂×#532	C₂⁵
LCS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ C₂×#62852 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

At least 27192 conjugacy classes of subgroups; only the 979 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=i,e^2=fg,f^2=kl,g^2=kl,j^2=l,b^a=bi,d^a=dh,e^a=ef,e^b=eg,e^c=efg,e^d=ejkl,f^a=fkl,f^b=fj,f^c=fkl,f^d=fkl,g^b=gjk,g^c=gkl,g^d=gk,h^e=hl,i^e=ij,i^f=ik,i^g=ikl,j^a=jk,j^b=jkl,j^c=jl,j^d=jl,j^i=jl,k^b=kl

Conjugacy classes & character table

139 conjugacy classes — view the full character table.

Relationships

automorphism group of

#7500, #7501, #7506, …

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