#72225495

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225494
commutator	C₈×C₃₂	C₂²×C₄
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	C₂³	#151659
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 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:

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

Polycyclic presentation:

2^12|a^2=e,b^2=ehjk,c^2=fhj,d^2=gij,e^2=l,f^2=hj,g^2=ikl,h^2=j,i^2=kl,k^2=l,b^a=bd,c^a=cgjl,c^b=cf,d^a=dg,d^b=dgjkl,e^b=ejl,e^d=el,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550, #72227136, #72227155
maximal normal subgroup of	#72225550, #72227136, #72227155

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