#72227279

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227278
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	C₂³	#140467
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

2^12|a^2=f,b^2=d,c^2=dhik,d^2=g,e^2=hil,f^2=ijk,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bfijk,c^a=ce,c^b=cfhijl,d^a=dkl,d^c=djkl,e^a=eh,e^b=ei,e^c=ehil,f^b=fijkl,f^c=fijkl,f^d=fl,g^a=gl,h^a=hj,h^b=hl,h^c=hjl,i^a=il,i^b=ik,i^c=ikl,j^a=jl,j^c=jl,k^b=kl,k^c=kl

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227291
maximal normal subgroup of	#72227291

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