#72227285

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	#72227284
commutator	C₁₆²	C₂²×C₈
Frattini, cosocle	#72216220	C₂³
prime radical	#72216368	C₄
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 ▷ #72216220 ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1954 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:

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

Polycyclic presentation:

2^13|a^2=d,c^2=hiklm,d^2=f,e^2=ghijl,f^2=ik,g^2=im,h^2=jk,i^2=km,j^2=l,k^2=m,l^2=m,b^a=be,c^a=chik,c^b=chijk,d^b=dg,d^c=dl,e^a=eh,e^b=egh,e^d=ekl,f^b=fi,f^c=fm,g^a=gkl,g^b=gikm,g^d=gm,h^a=hj,h^b=hjkl,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.8*F.9*F.11*F.12*F.13);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10*F.12);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.9*F.13);;
gap> SetPower(coll, 8, F.10*F.11);;
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.8*F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.13);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11*F.12);;
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);