#72227168

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	#72225571	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

2535 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:

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

Polycyclic presentation:

2^12|a^2=g,d^2=e,e^2=h,f^2=ik,g^2=jk,i^2=kl,k^2=l,b^a=bf,c^a=cgi,c^b=cfk,d^a=dijkl,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.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, 1, F.3*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11*F.12);;
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);