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#72227154

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72225202
commutatorC162C22×C4
Frattini, cosocleC2×C162C23
prime radicalC2×#123403C22
LCSG ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G
LECSG ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1

1241 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 155 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) = 512

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

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

Polycyclic presentation:

2^12|a^2=f,c^2=fikl,d^2=gi,e^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=l,b^a=bk,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dgk,e^a=ehk,e^b=eh,e^c=ehl,f^c=fk,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227169
maximal normal subgroup of#72227169
maximal quotient of#72227155

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