#72227267

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	#491465	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

840 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:

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

Polycyclic presentation:

2^12|a^2=d,b^2=dfg,c^2=dfghijl,d^2=g,e^2=hil,f^2=i,g^2=k,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=ce,c^b=cfhk,d^b=di,d^c=dijk,e^a=eh,e^b=eijk,e^c=ehil,f^a=fkl,f^b=fil,f^c=fil,f^d=fl,g^b=gk,g^c=gk,h^a=hj,h^c=hjl,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.4);;
gap> SetPower(coll, 2, F.4*F.6*F.7);;
gap> SetPower(coll, 3, F.4*F.6*F.7*F.8*F.9*F.10*F.12);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9*F.12);;
gap> SetPower(coll, 6, F.9);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
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);;
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);