#72227250

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	C₂×#151634	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

1421 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:

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

Polycyclic presentation:

2^12|a^2=d,d^2=g,e^2=hi,f^2=i,g^2=kl,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=ce,c^b=cfk,d^b=di,d^c=dijkl,e^a=eh,e^b=eil,e^c=ehijkl,f^a=fkl,f^b=fikl,f^c=fikl,f^d=fl,g^b=gk,g^c=gk,h^a=hj,h^b=hl,h^c=hj,i^a=il,i^b=ikl,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.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.11*F.12);;
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.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.11*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, 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*F.12);;
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);