#72227242

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72214696
commutator	C₈²	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72216582
prime radical	#491465	C₂×C₄
LCS	G ▷ C₈² ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216161 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1280 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:

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

Polycyclic presentation:

2^12|a^2=d,b^2=f,c^2=defgil,d^2=g,e^2=hijkl,f^2=ijl,g^2=jl,h^2=j,i^2=kl,j^2=l,k^2=l,b^a=be,c^a=chl,c^b=chijl,d^b=dh,d^c=djl,e^a=ei,e^b=ehjk,e^c=ejkl,f^a=fijl,f^c=fk,f^d=fl,g^b=gj,g^c=gl,h^b=hj,h^c=hl,i^a=ik,i^b=il,i^c=il,j^b=jl,k^a=kl

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227291
maximal normal subgroup of	#72227291

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