#72238422

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72236880
commutator	C₂³×C₂²⋊C₄	C₂×C₄²
Frattini, cosocle	C₂×#34980	C₂³
prime radical	C₂×#7670203	C₂²
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

At least 11495 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=f,c^2=g,d^2=jk,f^2=l,g^2=jl,b^a=be,c^a=cehkl,c^b=cd,d^a=djkl,d^b=di,d^c=dh,e^b=ejl,e^c=ejk,f^a=fjl,f^c=fijk,f^d=fjk,f^e=fl,g^a=gjl,g^b=ghjkl,g^d=gk,g^e=gl,g^f=gj,h^b=hj,h^c=hk,h^d=hl,h^f=hl,h^g=hl,i^b=ijk,i^c=ijl,j^b=jl,k^b=kl,k^c=kl

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238442
maximal normal subgroup of	#72238442

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