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#72227114

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72225182
commutatorC162C22×C4
Frattini, cosocleC2×C162C23
prime radicalC2×D162C2
LCSG ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G
LECSG ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1

2803 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 2803 conjugacy classes of subgroups — too many to draw. Only its 199 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:

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

Polycyclic presentation:

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

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227169
maximal normal subgroup of#72227169
maximal quotient of#72227115

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