#72227121

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	#72227120
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	C₂×#508	#106
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216211 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1805 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:

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

Polycyclic presentation:

2^12|a^2=e,b^2=eik,c^2=egj,d^2=gikl,e^2=h,f^2=il,g^2=j,h^2=l,i^2=k,j^2=l,k^2=l,b^a=bf,c^a=cg,c^b=cfgik,d^a=dijkl,d^b=dijkl,d^c=dgikl,e^b=ek,e^c=ej,f^a=fi,f^b=fi,f^c=fi,f^e=fl,g^a=gl,g^b=gl,g^c=gj,h^b=hl,h^c=hl,i^a=ik,i^b=ik,i^c=ik,j^c=jl,k^a=kl,k^b=kl,k^c=kl

Conjugacy classes & character table

208 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, 2, F.5*F.9*F.11);;
gap> SetPower(coll, 3, F.5*F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.9*F.11*F.12);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.12);;
gap> SetPower(coll, 7, F.10);;
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.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*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);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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);;
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);