#72256975

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#6453969
commutator, Frattini, cosocle	C₂³×C₈	C₂⁶
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #72239129 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

At least 33186 conjugacy classes of subgroups; only the 4965 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|c^2=i,d^2=i,e^2=g,f^2=gl,g^2=k,k^2=l,e^a=ej,e^c=eg,e^d=ek,f^a=fijl,f^b=fh,f^c=fg,f^d=fk,g^c=gk,g^d=gl,i^e=il,i^f=il,j^f=jl,k^c=kl

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

automorphism group of

#2158, #2249, C₂×#508

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