#72227126

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72225188
commutator	C₁₆²	C₂²×C₄
Frattini, cosocle	C₂×C₁₆²	C₂³
prime radical	#72225450	C₂
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^12|b^2=f,d^2=gik,e^2=hjl,g^2=ik,h^2=jl,i^2=k,j^2=l,b^a=bl,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dgk,e^a=ehl,e^b=eh,e^c=ehl,f^c=fk,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227169
maximal normal subgroup of	#72227169
maximal quotient of	#72227127

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