#72226924

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226923
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#72225576	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|c^2=e,d^2=gil,e^2=hj,f^2=h,g^2=il,h^2=j,i^2=kl,k^2=l,c^a=cg,c^b=cf,d^a=dgk,d^b=dgk,d^c=dk,e^a=ei,e^b=eh,e^d=el,f^a=fikl,f^b=fhj,g^a=gikl,g^b=gikl,g^c=gl,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982
maximal normal subgroup of	#72226982

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