#72226858

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	#72226857
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#72216368	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2078 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:

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

Polycyclic presentation:

2^12|a^2=e,c^2=fh,d^2=gijl,e^2=l,f^2=hj,g^2=ik,h^2=j,i^2=kl,k^2=l,b^a=bj,c^a=cg,c^b=cf,d^a=dgjkl,d^b=dgjl,e^b=el,e^d=el,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982, #72227143, #72227155
maximal normal subgroup of	#72226982, #72227143, #72227155

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