#72258177

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72225332
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72258176
prime radical	C₂³	#129173
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ #72225138 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=d,b^2=dij,c^2=dehj,d^2=g,e^2=h,f^2=ijkl,g^2=l,h^2=j,i^2=k,k^2=l,b^a=bf,c^a=cfhijl,c^b=cfh,d^b=dkl,d^c=dj,e^a=ei,e^b=eil,e^c=ej,f^a=fijk,f^b=fijk,f^c=fk,f^d=fl,g^b=gl,h^a=hk,h^b=hk,i^a=ikl,i^b=ikl,i^c=il,j^a=jl,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258194
maximal normal subgroup of	#72258194

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