#72227077

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227065
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	C₂×#151634	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216215 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1969 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:

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

Polycyclic presentation:

2^12|a^2=e,d^2=g,e^2=h,f^2=il,g^2=jkl,h^2=l,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=cg,c^b=cfil,d^a=dij,d^b=dikl,d^c=dgjkl,e^b=ek,e^c=ej,f^a=fi,f^b=fik,f^c=fik,f^e=fl,g^a=gkl,g^b=gkl,g^c=gjkl,h^b=hl,h^c=hl,i^a=ik,i^b=ikl,i^c=ikl,j^c=jl,k^a=kl,k^b=kl,k^c=kl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227231
maximal normal subgroup of	#72227231

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