#72287177

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#93757
commutator	C₂×C₃₂	C₂²×C₁₆
Frattini, cosocle	C₂×#7330	C₂³
prime radical	#512934	C₈
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#1311 ◁ C₂²×#7330 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4);(6,12,14,30,15,31,41,18,17,33,27,49,43,34,63,50)(7,13,19,36,20,37,51,39)(8,22,24,55)(9,26)(10,29)(16,44,46,32,23,57,54,64,47,66,42,65,59,35,60,58)(21,53,45,38,52,68,67,40)(25,62,61,56)(28,48);(2,4)(5,6,13,35,56,42,7,18,48,41,68,64,25,47,67,33,29,43,36,44,22,54,51,49,28,14,40,65,8,23,52,34,11,17,37,57,62,46,20,50,26,63,53,58,61,16,45,12,10,15,39,66,55,60,19,30,9,27,38,32,24,59,21,31)

Polycyclic presentation:

2^12|b^2=d,c^2=deghl,d^2=g,e^2=h,g^2=i,h^2=j,j^2=k,k^2=l,c^a=cf,c^b=ce,d^c=djkl,e^b=eh,e^c=ej,e^d=ek,g^c=gkl,g^e=gl,h^b=hj,h^c=hk,h^d=hl,i^c=il,j^b=jk,j^c=jl,k^b=kl

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287333, #72323191
maximal normal subgroup of	#72287333, #72323191

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