#72225124

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216199
commutator	C₁₆²	C₂³×C₄
Frattini, cosocle	C₂×C₁₆²	C₂⁴
prime radical	#72225108	C₂
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216182 ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

4776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^13|c^2=fi,d^2=g,e^2=hj,f^2=ik,h^2=jl,i^2=km,j^2=l,k^2=m,b^a=be,c^a=cf,c^b=cf,d^a=dl,d^b=djl,d^c=dk,e^a=eh,e^b=eh,e^d=el,f^a=fi,f^b=fi,f^d=fm,g^b=gl,g^c=gm,h^a=hj,h^b=hj,i^a=ik,i^b=ik,j^a=jl,j^b=jl,k^a=km,k^b=km

Conjugacy classes & character table

344 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225329, #72227170
maximal normal subgroup of	#72225329, #72227170

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11*F.13);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.12);;
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, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.13);;
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> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);