#72225115

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₃₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216196
commutator	C₁₆²	C₂²×C₄
Frattini, cosocle	C₂×C₁₆²	C₂³
prime radical	#72216368	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

1521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^12|c^2=efj,d^2=gi,e^2=hj,f^2=jl,g^2=ik,h^2=jl,i^2=k,j^2=l,b^a=bd,c^a=ce,c^b=cei,d^a=dg,d^b=dg,d^c=dk,e^a=eh,e^b=eh,e^c=el,f^a=fj,f^b=fjk,f^c=fl,g^a=gi,g^b=gi,h^a=hj,h^b=hj,i^a=ik,i^b=ik,j^a=jl,j^b=jl

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225124, #72227115, #72227155
maximal normal subgroup of	#72225124, #72227115, #72227155

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