Finite Group Databasebeta

← Prev (#72225122) Next (#72225124) →

#72225123

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C16

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72216197
commutatorC162C22×C4
Frattini, cosocleC2×C162C23
prime radicalC23#123411
LCSG ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×C162 ◁ G
LECSG ▷ C2×C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1

933 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 933 conjugacy classes of subgroups — too many to draw. Only its 427 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)(9,26,25,32)(10,27,20,24)(12,22,30,17)(13,19)(14,31)(15,29)(21,28)(34,39,41,52)(35,44)(36,48)(37,49)(40,46)(42,59,57,55)(43,61,58,54)(45,50)(47,63)(51,64,60,53)(56,62);(1,2,5,20,6,8,18,10)(3,12,14,32,15,30,28,26)(4,17,16,9,19,22,13,25)(7,21,27,29,23,31,24,11)(34,39,41,52)(35,44)(36,48)(37,49)(40,46)(42,59,57,55)(43,61,58,54)(45,50)(47,63)(51,64,60,53)(56,62);(33,34,40,55,36,42,47,52,49,58,63,64,45,51,46,54,38,41,56,59,50,57,35,39,37,43,44,53,48,60,62,61)

Polycyclic presentation:

2^12|a^2=f,b^2=fi,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=ce,d^a=dg,d^b=dg,e^a=eh,e^b=eh,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

460 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225124, #72227136
maximal normal subgroup of#72225124, #72227136

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, 2, F.6*F.9);;
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);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
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);