#72235339

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	#13827
commutator	C₂²×C₄²	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄²	C₂⁴
socle	C₂²	#72235154
prime radical	C₂⁶	H(3,𝔽₄)
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄² ▷ C₂² ▷ C₁

46163 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=ehik,d^2=fhij,h^2=kl,i^2=l,j^2=l,b^a=bg,c^a=ci,c^b=ch,d^a=dhijl,d^b=dhikl,d^c=dgij,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,i^a=ikl,i^b=il,i^c=ik,i^d=ikl

Conjugacy classes & character table

424 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72235394
Fitting subgroup of	#72235394
maximal subgroup of	#72235394
maximal normal subgroup of	#72235394

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