#72227195

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#129088
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×M₆₄	C₂³
socle	C₂	#72227194
prime radical	C₂³	#151045
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #1397 ◁ #149495 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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:

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

Polycyclic presentation:

2^12|a^2=d,b^2=eij,c^2=f,d^2=g,e^2=hl,f^2=ijl,g^2=i,h^2=j,i^2=k,j^2=l,k^2=l,b^a=be,c^a=ci,c^b=cl,d^b=dj,d^c=dkl,e^a=eh,e^d=el,f^a=fk,f^d=fl,g^b=gl,g^c=gl,h^a=hj,j^a=jl

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227218, #72227257, #72227285
maximal normal subgroup of	#72227218, #72227257, #72227285

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