#72227014

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#94048
commutator	C₂×C₃₂	C₄×C₁₆
Frattini, cosocle	C₁₆×M₆₄	C₂²
socle	C₂	#72227013
prime radical	C₂³	#93752
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₃₂ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=e,b^2=c,c^2=f,d^2=gkl,e^2=l,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bd,c^a=cg,d^a=dgij,d^b=dk,d^c=dl,e^b=ej,e^d=el,f^a=fi,g^a=gikl,g^b=gl,h^a=hk,i^a=ikl,j^a=jl,k^a=kl

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227029, #72227127, #72227155
maximal normal subgroup of	#72227029, #72227127, #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.3);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.11*F.12);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);