#72258200

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72225244
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	#72216275	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72225535 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,3,9,5,7,6,11)(4,13,12,10,15,8,14,16)(18,24,20,32)(19,28,30,47,29,45,39,41)(21,35)(22,38)(25,43,44,48,33,37,46,31)(26,40)(27,36,34,42);(2,7)(3,6)(4,14)(8,16)(10,13)(12,15)(17,18)(19,29)(20,23)(21,36)(22,34)(24,40)(25,28)(26,42)(27,38)(31,43)(32,35)(33,45)(37,48)(41,44)(46,47);(2,8)(3,6)(4,14)(7,13)(9,10)(11,16)(12,15)(17,19)(18,25)(20,33)(21,37)(22,39)(23,29)(24,41)(26,43)(27,44)(28,42)(30,38)(31,40)(32,47)(34,46)(35,48)(36,45)

Polycyclic presentation:

2^12|a^2=d,d^2=g,e^2=h,f^2=ik,h^2=j,i^2=k,j^2=l,b^a=bhi,c^a=ce,c^b=cefi,d^b=dkl,d^c=djl,e^a=eh,e^b=ehjl,e^c=ehjl,e^d=el,f^a=fk,f^b=fi,f^c=fi,g^c=gl,h^a=hj,h^b=hjl,h^c=hjl,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

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