#72227232

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72214397
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72225250
prime radical	#72216236	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1613 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:

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

Polycyclic presentation:

2^12|a^2=d,c^2=eijk,d^2=g,e^2=hijkl,f^2=ijk,g^2=k,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=be,c^a=cefij,c^b=cehjk,d^b=di,d^c=djk,e^a=eh,e^b=ehi,f^a=fkl,f^b=fij,f^d=fl,g^b=gk,h^a=hj,h^b=hj,i^b=ikl,j^a=jl,j^b=jl,k^b=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227244
maximal normal subgroup of	#72227244

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