#72287138

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	#72224639
commutator	C₂×C₃₂	C₂²×C₁₆
Frattini, cosocle	C₂×#7330	C₂³
prime radical	C₂×#13742	C₈
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#93668 ◁ G
LECS	G ▷ C₂×#7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

845 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)(6,12,15,32,16,35,34,63,17,36,18,39,19,37,50,64)(7,20,22,53,24,54,55,67)(8,25,27,59)(9,29)(13,31,38,33,40,61,42,65,41,62,52,66,51,68,49,44)(14,43,45,47,46,58,21,48)(23,57,26,56)(28,60);(6,13,17,41)(7,21)(8,26)(9,28)(10,30)(12,31,36,62)(14,22)(15,38,18,52)(16,40,19,51)(20,48)(23,27)(24,45)(25,56)(29,60)(32,33,39,66)(34,42,50,49)(35,61,37,68)(43,53)(44,63,65,64)(46,55)(47,54)(57,59)(58,67);(2,4)(5,6,14,44,8,12,20,42,28,34,43,31,57,64,7,13,10,19,21,33,25,37,53,52,29,15,48,61,23,39,22,40,11,17,46,65,27,36,54,49,60,50,58,62,56,63,24,41,30,16,45,66,59,35,67,38,9,18,47,68,26,32,55,51)

Polycyclic presentation:

2^12|a^2=d,b^2=i,c^2=ehkl,d^2=g,e^2=hjl,f^2=jk,g^2=i,h^2=jk,j^2=kl,k^2=l,c^a=cf,c^b=ce,d^c=djkl,e^a=ej,e^b=eh,e^d=ek,f^a=fk,f^b=fj,f^d=fl,g^c=gk,g^e=gl,h^a=hk,h^b=hj,h^d=hl,i^c=il,j^a=jl,j^b=jk,k^b=kl

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287333
maximal normal subgroup of	#72287333

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