#72225534

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225533
commutator	C₈×C₃₂	C₂²×C₄
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	#72216261	C₂²
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ #72216226 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,14,6,13,30,50,36,17,33,11,34,15,22,47,9,18,29,48,7,37,5,32,8,16,31,49,35,56,28,64)(3,19,24,54,26,44,59,41,58,42,60,51,63,45,61,55,23,40,20,46,27,43,25,12,21,52,57,53,62,10,38,39);(2,10)(3,20)(4,28)(5,33)(6,35)(7,34)(8,36)(11,46)(12,50)(13,52)(14,53)(15,40)(16,44)(17,43)(18,45)(19,56)(21,57)(22,29)(23,24)(25,62)(26,61)(27,38)(30,31)(32,41)(37,42)(39,64)(47,55)(48,51)(49,54)(58,60)(59,63);(2,11,13,47,17,48,15,32,18,49,37,64,16,14,56,50)(3,21,23,58)(4,29)(5,30)(10,39,40,46,44,41,52,53,45,55,19,54,43,12,42,51)(20,57,24,60)(22,28)(25,38,59,61)(26,62,27,63)(31,33)

Polycyclic presentation:

2^12|a^2=f,c^2=e,d^2=ghi,e^2=hikl,f^2=h,g^2=ik,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bd,c^a=cgijk,c^b=cfgjl,d^a=dg,d^b=dghj,d^c=dk,e^a=eil,e^b=ehikl,e^d=el,f^b=fhj,f^c=fl,g^a=gi,g^b=gi,g^c=gl,h^b=hj,i^a=ik,i^b=ik,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550
maximal normal subgroup of	#72225550

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