#72227218

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227217
commutator	C₈×C₃₂	C₂²×C₈
Frattini, cosocle	C₈×#1398	C₂³
prime radical	#72216278	C₄
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×#346 ◁ #72216231 ◁ G
LECS	G ▷ C₈×#1398 ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^13|a^2=d,c^2=fi,d^2=g,e^2=hkm,f^2=ik,g^2=k,h^2=j,i^2=km,j^2=l,k^2=m,l^2=m,b^a=be,c^a=cijkl,c^b=cf,d^b=djklm,d^c=dk,e^a=eh,e^b=ehjklm,e^d=el,f^a=fklm,f^b=fi,f^d=fm,g^b=gl,g^c=gm,g^e=gm,h^a=hj,h^b=hjlm,h^d=hm,i^b=ik,j^a=jl,j^b=jlm,k^b=km,l^a=lm,l^b=lm

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

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