#72340467

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216433
commutator, Frattini, cosocle	C₂×H(7,𝔽₂)	C₂⁴
LCS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁵ ◁ C₂×H(7,𝔽₂) ◁ G
LECS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

Only the 225 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 225 normal subgroups are shown; non-normal subgroups are omitted.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|c^2=l,d^2=k,g^2=l,b^a=bf,c^a=ckl,c^b=ce,d^b=dg,d^c=dk,e^a=eh,e^d=ej,f^c=fhl,f^d=fi,g^a=gil,g^b=gl,g^c=gj,g^e=gl,g^f=gl,h^d=hk,h^g=hl,i^b=il,i^c=ik,i^e=il,j^a=jk,j^b=jl,j^f=jl,k^b=kl

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340477
maximal normal subgroup of	#72340477

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