#72293316

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72292810
commutator	#376	C₂³×C₈
Frattini, cosocle	C₂×#1746	C₂⁴
prime radical	C₂²×#45355	C₄
LCS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#20365 ◁ G
LECS	G ▷ C₂×#1746 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

58431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^12|c^2=e,d^2=jl,e^2=h,f^2=jl,g^2=kl,h^2=l,i^2=l,c^a=cg,c^b=cf,d^a=dj,d^b=djl,d^c=dil,e^a=ekl,e^b=ei,e^d=ekl,f^a=fj,f^b=fjl,f^c=fijl,f^e=fk,g^a=gkl,g^b=gjl,g^d=gl,g^f=gl,h^b=hl,i^b=il,i^c=ik,j^c=jl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

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