#72225523

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225522
commutator	C₄×C₃₂	C₂×C₄²
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	#72216261	C₂²
LCS	G ▷ C₄×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×#346 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

900 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:

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

Polycyclic presentation:

2^12|a^2=f,c^2=e,d^2=ghi,f^2=h,g^2=ik,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bd,c^a=cjl,c^b=chijl,d^a=dg,d^b=dghj,d^c=dk,e^a=el,e^b=ejkl,e^d=el,f^b=fhj,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

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550
maximal normal subgroup of	#72225550

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