#72227105

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	#72227104
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#72216275	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

2171 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:

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

Polycyclic presentation:

2^12|a^2=e,b^2=eik,d^2=g,e^2=h,f^2=il,g^2=jk,h^2=l,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=cg,c^b=cfgk,d^a=dijk,d^b=dijk,d^c=dgjk,e^b=ek,e^c=ej,f^a=fi,f^b=fi,f^c=fik,f^e=fl,g^a=gk,g^b=gk,g^c=gjk,h^b=hl,h^c=hl,i^a=ik,i^b=ik,i^c=ikl,j^c=jl,k^a=kl,k^b=kl,k^c=kl

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227231, #72227244
maximal normal subgroup of	#72227231, #72227244

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