#72324296

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72251779
commutator	C₂²×#123	C₂³×C₄
Frattini, cosocle	C₂²×#534	C₂⁴
prime radical	#72305966	C₂
LCS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#534 ◁ G
LECS	G ▷ C₂²×#534 ▷ C₂⁴ ▷ C₂² ▷ C₁

At least 19202 conjugacy classes of subgroups; only the 727 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=g,c^2=ijk,f^2=l,b^a=be,c^a=cf,d^a=dh,d^b=dj,d^c=dj,e^c=ei,e^d=ej,f^a=fjkl,f^b=fik,f^c=fk,f^d=fl,f^e=fk,g^c=gjk,g^d=gl,g^f=gl,h^a=hl,h^b=hj,h^e=hl,i^a=ik,j^a=jl

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72324301, #72324348, #72324755
maximal normal subgroup of	#72324301, #72324348, #72324755

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