#72227166

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	#72227165
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#72225567	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216234 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2538 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:

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

Polycyclic presentation:

2^12|a^2=l,b^2=g,d^2=e,e^2=h,f^2=ik,g^2=jk,i^2=kl,k^2=l,b^a=bf,c^b=cfgi,d^a=dik,d^b=dj,d^c=dgij,e^a=ekl,e^c=ej,f^a=fi,f^b=fi,f^c=fi,f^d=fk,f^e=fl,g^c=gjkl,g^d=gl,h^a=hl,i^a=ik,i^b=ik,i^c=ik,i^d=il,j^a=jl,j^b=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.12);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 9, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.11*F.12);;
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);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 10, 2, 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);