#72226950

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226949
commutator	C₈×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁴
prime radical	#72225583	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=f,c^2=e,d^2=gik,e^2=hj,f^2=hij,g^2=i,h^2=j,i^2=kl,k^2=l,b^a=bfgik,c^a=cgjkl,c^b=cfghijk,d^a=dg,d^b=dg,d^c=dk,e^a=ei,e^b=eh,e^d=el,f^b=fhik,f^c=fl,g^a=gik,g^b=gik,g^c=gl,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982
maximal normal subgroup of	#72226982

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