#72226963

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	#72226957
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#72225587	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

2176 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)(3,19)(4,25)(5,29)(6,12)(7,35)(8,16)(9,17)(10,21)(11,36)(13,31)(14,53)(15,55)(18,39)(20,44)(22,40)(23,58)(24,59)(26,56)(27,47)(28,43)(30,49)(32,57)(33,41)(34,50)(37,52)(38,45)(42,62)(46,60)(48,64)(51,63)(54,61);(1,3)(2,10)(4,26)(5,28)(6,33)(7,36)(8,38)(9,24)(11,29)(12,19)(13,45)(14,54)(15,40)(16,43)(17,25)(18,42)(20,31)(21,55)(22,53)(23,39)(27,30)(32,62)(34,64)(35,41)(37,60)(44,49)(46,57)(47,56)(48,52)(50,59)(51,58)(61,63);(1,4,5,23,30,33,34,27,39,62,55,3,7,20,31,60,9,21,32,38,53,64,6,22,8,28,26,24,37,61,63,36)(2,11,13,44,52,59,15,42,16,40,50,41,14,45,29,25,17,46,51,54,35,19,56,43,18,47,12,48,49,58,57,10);(2,12,35,29,16,13,49,56,17,50,52,57,18,51,14,15)(10,40,54,58,42,46,48,59,25,47,44,45,43,11,41,19)

Polycyclic presentation:

2^12|c^2=e,d^2=fh,e^2=ik,f^2=hj,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bg,c^a=cgk,c^b=cgjk,d^a=dg,d^b=df,d^c=dj,e^a=ei,e^b=ei,f^a=fikl,f^b=fh,g^a=gikl,g^b=gikl,g^c=gl,h^a=hk,h^b=hj,i^a=ikl,i^b=ikl,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982
maximal normal subgroup of	#72226982

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