#72227291

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227290
commutator	C₈×C₁₆	C₂³×C₈
Frattini, cosocle	C₈×#346	C₂⁴
prime radical	#72225583	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216161 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3000 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:

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

Polycyclic presentation:

2^13|a^2=e,d^2=eghj,e^2=h,f^2=ij,g^2=j,h^2=lm,i^2=km,j^2=l,k^2=m,l^2=m,b^a=bg,c^a=cf,c^b=cgl,d^a=djlm,d^c=dijklm,e^b=ej,e^c=ejklm,e^d=elm,f^a=fi,f^b=fjm,f^c=fijklm,f^d=fklm,g^a=glm,g^b=gjlm,g^c=gjlm,g^d=gl,g^e=gm,h^b=hl,h^c=hl,h^d=hm,i^a=ik,i^b=im,i^c=ik,i^d=im,j^a=jm,j^b=jlm,j^c=jlm,j^d=jm,k^a=km,k^c=km,l^b=lm,l^c=lm

Conjugacy classes & character table

182 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.5);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.12*F.13);;
gap> SetPower(coll, 9, F.11*F.13);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.12*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.12*F.13);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 6, 4, F.6*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 1, F.7*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.13);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.10*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);