#72227175

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	#72227174
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#72225569	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

1830 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:

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

Polycyclic presentation:

2^12|a^2=g,b^2=gl,d^2=e,e^2=h,f^2=ik,g^2=jkl,i^2=kl,k^2=l,b^a=bf,c^a=cgikl,c^b=cfgl,d^a=dij,d^b=dj,d^c=dgijkl,e^a=ek,e^c=ej,f^a=fi,f^b=fi,f^c=fi,f^d=fk,f^e=fl,g^c=gjk,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

148 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, 2, F.7*F.12);;
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*F.12);;
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*F.11*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, 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, 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);;
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);