#72227059

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227058
commutator	C₁₆²	C₂×C₄²
Frattini, cosocle	C₁₆×M₆₄	C₂³
prime radical	#72216368	C₂²
LCS	G ▷ C₁₆² ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×#346 ◁ #72216231 ◁ G
LECS	G ▷ C₁₆×M₆₄ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1966 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:

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

Polycyclic presentation:

2^13|a^2=e,c^2=f,d^2=ghi,e^2=h,g^2=ik,h^2=jl,i^2=km,j^2=lm,k^2=m,l^2=m,b^a=bd,c^a=cij,c^b=cklm,d^a=dg,d^b=dghj,d^c=dklm,e^b=ehj,e^c=el,f^a=fkl,f^e=fm,g^a=gi,g^b=gi,g^c=gm,h^b=hj,h^c=hm,i^a=ik,i^b=ik,j^b=jl,k^a=km,k^b=km,l^b=lm

Conjugacy classes & character table

188 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.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, 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.13);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 11, F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.11*F.12*F.13);;
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*F.12*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
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.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
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.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);