#72226838

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	#72226836
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#33981	D₈
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

1580 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)(3,16)(4,9)(5,11)(6,8)(7,14)(10,20)(12,18)(13,27)(15,28)(17,31)(19,44)(21,45)(22,47)(23,37)(24,35)(25,30)(26,38)(29,33)(32,54)(34,61)(36,57)(39,60)(40,52)(41,58)(42,51)(43,50)(46,53)(48,59)(49,56)(55,63)(62,64);(2,8)(3,17,19,50)(4,23)(5,27)(6,28)(9,34)(10,16,40,44)(11,33)(12,39)(14,15)(18,26)(20,56,52,22)(21,53,55,58)(24,29)(25,54)(30,37)(31,49,43,47)(32,38)(36,41,48,46)(42,45,62,63)(51,57,64,59)(60,61);(2,9,35,32,8,30,11,37,14,38,13,34,15,39,33,12)(10,36,62,47,31,45,41,44,40,48,42,49,43,63,46,16);(1,3,18,53,29,55,60,50,28,56,61,51,27,59,26,52,7,19,23,58,5,21,25,17,6,22,54,64,24,57,4,20)(2,10,9,36,35,62,32,47,8,31,30,45,11,41,37,44,14,40,38,48,13,42,34,49,15,43,39,63,33,46,12,16)

Polycyclic presentation:

2^12|b^2=e,c^2=fhj,d^2=gil,f^2=hj,g^2=ikl,h^2=j,i^2=kl,k^2=l,b^a=bk,c^a=cg,c^b=cf,d^a=dg,d^b=dgk,e^d=el,f^a=fi,f^b=fh,g^a=gil,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

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982, #72227127, #72227136
maximal normal subgroup of	#72226982, #72227127, #72227136

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