#72258194

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72225335
commutator	C₄×C₁₆	C₂²×C₄×C₈
Frattini, cosocle	C₈×#346	C₂⁴
socle	C₂	#72258190
prime radical	#33981	#97
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#1398 ◁ #72225206 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^13|a^2=m,c^2=e,d^2=efik,e^2=h,f^2=i,g^2=jlm,h^2=k,i^2=k,j^2=l,l^2=m,b^a=bg,c^a=cg,c^b=cgj,d^a=dj,d^c=di,e^a=ej,e^b=ejlm,e^d=ek,f^a=fjl,f^b=fjlm,f^c=fk,f^d=fk,g^a=gj,g^b=gj,g^c=gl,g^d=glm,g^e=gm,h^a=hl,h^b=hlm,i^a=ilm,i^b=ilm,j^a=jlm,j^b=jlm,j^c=jm,j^d=jm,k^a=km,k^b=km,l^a=lm,l^b=lm

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

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