#72227270

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, socle	C₂	#72227249
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	#72225339	C₄
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

1338 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:

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

Polycyclic presentation:

2^12|a^2=f,c^2=d,d^2=g,e^2=hijk,f^2=il,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bf,c^a=ce,c^b=cfkl,d^a=djk,d^b=dk,e^a=eh,e^b=ei,e^c=ehijk,f^b=fikl,f^c=fi,f^d=fl,g^b=gl,h^a=hj,h^c=hjl,i^b=ik,i^c=ikl,j^a=jl,j^c=jl,k^b=kl,k^c=kl

Conjugacy classes & character table

160 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, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.9*F.12);;
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);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.11*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
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.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*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);