#72227275

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227266
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	C₂×#12515	C₂×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

923 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:

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

Polycyclic presentation:

2^12|a^2=f,b^2=d,d^2=g,e^2=hil,f^2=ij,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bfij,c^a=ce,c^b=cfhij,d^a=dk,d^c=djk,e^a=eh,e^b=ei,e^c=ehijkl,f^b=fijl,f^c=fijkl,f^d=fl,g^a=gl,h^a=hj,h^b=hl,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^c=jl,k^b=kl,k^c=kl

Conjugacy classes & character table

124 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, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9*F.12);;
gap> SetPower(coll, 6, F.9*F.10);;
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);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
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.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*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);;
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);;
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);