#72226993

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	#72226991
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#33981	C₄⋊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

1825 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:

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

Polycyclic presentation:

2^12|a^2=l,c^2=e,d^2=g,e^2=h,f^2=ik,g^2=j,i^2=kl,k^2=l,b^a=bf,c^a=ci,c^b=cfikl,d^a=di,d^b=di,d^c=dg,e^a=el,e^b=ek,f^a=fi,f^b=fi,f^c=fikl,f^e=fl,g^a=gkl,g^b=gkl,g^c=gj,h^b=hl,i^a=ik,i^b=ik,i^c=ikl,j^a=jl,j^b=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.12);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.10);;
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, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
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*F.12);;
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);;
gap> SetConjugate(coll, 8, 2, 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, 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);