Finite Group Databasebeta

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#72225119

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C32

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72216196
commutatorC162C22×C4
Frattini, cosocleC2×C162C23
prime radical#72216278C2
LCSG ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G
LECSG ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1

1365 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 1365 conjugacy classes of subgroups — too many to draw. Only its 295 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 1024

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=f,c^2=ehjl,d^2=gik,e^2=hjl,g^2=ik,h^2=jl,i^2=k,j^2=l,b^a=bd,c^a=ce,c^b=cejl,d^a=dg,d^b=dgk,e^a=eh,e^b=ehl,f^b=fk,f^c=fl,g^a=gi,g^b=gi,h^a=hj,h^b=hj,i^a=ik,i^b=ik,j^a=jl,j^b=jl

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225124, #72227127
maximal normal subgroup of#72225124, #72227127

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