#72227136

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227135
commutator	C₁₆×C₃₂	C₂²×C₄
Frattini, cosocle	C₁₆×M₆₄	C₂³
prime radical	#33981	SD₃₂
LCS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1196 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:

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

Polycyclic presentation:

2^13|a^2=m,b^2=f,c^2=fi,d^2=gik,e^2=hjl,g^2=ik,h^2=jlm,i^2=k,j^2=lm,l^2=m,b^a=bl,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dg,e^a=ehl,e^b=ehm,e^c=ehm,f^c=fk,f^e=fm,g^a=gj,g^b=gi,g^c=gi,h^a=hjm,h^b=hj,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl,k^a=km,l^a=lm,l^b=lm,l^c=lm

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227170
maximal normal subgroup of	#72227170

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.13);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
gap> SetPower(coll, 5, F.8*F.10*F.12);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12*F.13);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);