#72227244

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	#72227243
commutator	C₈×C₁₆	C₂³×C₈
Frattini, cosocle	C₈×#346	C₂⁴
prime radical	#72216368	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3232 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:

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

Polycyclic presentation:

2^13|a^2=e,c^2=fgjkm,d^2=gi,e^2=h,f^2=ijklm,g^2=jk,h^2=lm,i^2=km,j^2=l,k^2=m,l^2=m,b^a=bf,c^a=cfgjkl,c^b=cfgijlm,d^a=dijkm,d^b=dgijlm,e^b=ej,e^c=ekm,e^d=elm,f^a=fi,f^b=fij,g^a=gl,g^b=gjkl,g^e=gm,h^b=hl,h^c=hm,h^d=hm,i^a=ik,i^b=ik,j^b=jlm,k^a=km,k^b=km,l^b=lm

Conjugacy classes & character table

212 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, 3, F.6*F.7*F.10*F.11*F.13);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10*F.11*F.12*F.13);;
gap> SetPower(coll, 7, F.10*F.11);;
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.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11*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.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11*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.13);;
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.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 11, 2, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);