#72225181

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216171
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	#72216236	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216182 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^12|a^2=d,c^2=f,d^2=g,e^2=h,f^2=i,g^2=k,h^2=j,i^2=k,j^2=l,b^a=be,c^a=cf,c^b=cfik,d^b=djl,d^c=dk,e^a=eh,e^b=ehjl,e^d=el,f^a=fi,f^b=fik,g^b=gl,h^a=hj,h^b=hjl,i^a=ik,i^b=ik,j^a=jl,j^b=jl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225192, #72225268, #72225300
maximal normal subgroup of	#72225192, #72225268, #72225300

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