#72307010

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#72213671
commutator, Frattini, cosocle	#3151	C₂⁵
LCS	G ▷ #3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ #10575509 ◁ G
LECS	G ▷ #3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

At least 24552 conjugacy classes of subgroups; only the 993 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=l,b^2=ikl,d^2=ikl,e^2=fijkl,f^2=jl,i^2=k,j^2=l,b^a=bikl,c^a=ck,d^a=dikl,e^a=eg,e^b=eh,e^c=eghik,e^d=ef,f^a=fi,f^b=fijl,f^c=fkl,f^d=fl,f^e=fjl,g^b=gjl,g^c=gl,g^d=gikl,g^e=gik,g^f=gj,h^a=hk,h^b=hj,h^c=hl,h^d=hil,h^e=hijl,h^f=hjl,h^g=hk,i^a=ik,i^e=ij,i^g=ik,i^h=ik,j^a=jk,j^b=jkl,j^e=jl,j^g=jl,j^h=jl,k^e=kl

Conjugacy classes & character table

142 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, 1, F.12);;
gap> SetPower(coll, 2, F.9*F.11*F.12);;
gap> SetPower(coll, 4, F.9*F.11*F.12);;
gap> SetPower(coll, 5, F.6*F.9*F.10*F.11*F.12);;
gap> SetPower(coll, 6, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.11*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.12);;
gap> SetConjugate(coll, 11, 5, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);