#72227239

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72225332
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72227238
prime radical	C₂³	#129125
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ #72216161 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^12|a^2=d,b^2=f,c^2=dfgl,d^2=g,e^2=hk,f^2=i,g^2=k,h^2=j,i^2=k,j^2=l,k^2=l,b^a=be,c^a=cil,c^b=chijkl,d^b=djk,d^c=dk,e^a=eh,e^b=ehj,e^c=ejl,e^d=el,f^a=fkl,f^c=fk,f^d=fl,g^c=gl,h^a=hj,h^b=hjl,h^c=hl,i^a=il,i^c=il,j^a=jl,j^b=jl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227291, #72258194
maximal normal subgroup of	#72227291, #72258194

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