#72226936

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226935
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#72225583	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2014 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:

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

Polycyclic presentation:

2^12|a^2=f,c^2=e,d^2=gi,f^2=hj,g^2=ik,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bfgi,c^a=cj,c^b=cjkl,d^a=dg,d^b=dg,d^c=dkl,e^a=el,e^d=el,f^b=fh,f^c=fl,g^a=gi,g^b=gi,g^c=gl,h^b=hj,i^a=ik,i^b=ik,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982, #72227291
maximal normal subgroup of	#72226982, #72227291

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