#72216408

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#7822
commutator	C₂×#1744	C₂²×C₄
Frattini, cosocle	C₂²×#1744	C₂³
socle	C₂³	#97534
prime radical	#72213466	C₂
LCS	G ▷ C₂×#1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂²×#1744 ◁ G
LECS	G ▷ C₂²×#1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

13038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|a^2=e,b^2=i,c^2=d,d^2=jl,f^2=hik,g^2=j,h^2=k,i^2=l,b^a=beil,c^a=cf,c^b=cg,d^a=dh,d^b=dj,f^a=fhil,f^b=fhl,f^c=fikl,f^d=fl,g^a=gh,g^b=gj,g^f=gl,h^a=hk,h^b=hk,h^c=hl,i^a=il,j^a=jk

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of

#72218147

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