#72235722

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#17081
commutator	C₂²×C₄²	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄²	C₂⁴
socle	C₂²	#72235525
prime radical	C₂⁶	H(3,𝔽₄)
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄² ▷ C₂² ▷ C₁

44795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^12|a^2=e,b^2=f,c^2=fgh,d^2=efgijl,e^2=k,f^2=kl,h^2=kl,i^2=kl,j^2=k,b^a=bg,c^a=ci,c^b=cj,d^a=dgj,d^b=dghjk,d^c=dghil,e^b=ek,e^c=ek,f^a=fkl,f^c=fkl,f^d=fkl,g^a=gk,g^b=gkl,g^c=gl,g^d=gk,h^a=hk,h^c=hk,h^d=hk,i^a=il,i^b=il,i^d=il,j^b=jl,j^c=jl

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72235731
Fitting subgroup of	#72235731
maximal subgroup of	#72235731
maximal normal subgroup of	#72235731

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