#72227091

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227034
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#72225394	C₄
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

3807 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:

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

Polycyclic presentation:

2^12|a^2=e,c^2=egj,d^2=gil,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=cfik,d^a=dijl,d^b=dikl,d^c=dgil,e^b=ek,e^c=ej,f^a=fi,f^b=fik,f^c=fi,f^e=fl,g^a=gl,g^c=gj,h^b=hl,h^c=hl,i^a=ik,i^b=ikl,i^c=ik,j^c=jl,k^a=kl,k^b=kl,k^c=kl

Conjugacy classes & character table

196 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, 3, F.5*F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.9*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.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*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);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, 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*F.12);;
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);