#72225464

order	4096
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	#72225463
commutator	C₈×C₃₂	C₂²×C₄
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	#33981	Q₁₆
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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)(3,8,19,35)(4,9,26,40)(5,27,29,12)(6,11,28,16)(10,42,47,20)(13,23,38,18)(14,50,48,24)(17,45,52,44)(21,51,53,31)(22,33,58,30)(25,37,56,41)(32,60,36,55)(34,63,49,59)(39,61,43,57)(46,64,62,54);(2,8,11,30,15,35,16,33)(3,17,19,52)(4,23)(5,24)(6,28)(9,36,37,46,40,32,41,62)(10,43,44,13,47,39,45,38)(12,49,48,31,27,34,14,51)(18,26)(20,58,42,22)(21,60,53,55)(25,57)(29,50)(54,63,64,59)(56,61);(2,9,14,39,11,37,12,38,15,40,48,43,16,41,27,13)(8,10,34,46,33,45,31,36,35,47,49,62,30,44,51,32)

Polycyclic presentation:

2^12|a^2=l,b^2=e,c^2=fhj,d^2=gi,e^2=j,f^2=hj,g^2=ik,h^2=j,i^2=kl,k^2=l,b^a=bd,c^a=cg,c^b=cf,d^a=dg,d^b=dgk,e^a=ek,e^d=el,f^a=fil,f^b=fh,g^a=gi,g^b=gil,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

238 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550, #72227115, #72227136
maximal normal subgroup of	#72225550, #72227115, #72227136

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