Finite Group Databasebeta

← Prev (#72340932) Next (#72340934) →

#72340933

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20585C23
prime radical#72285341C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2127 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2127 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

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

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

Polycyclic presentation:

2^11|a^2=d,b^2=ghk,c^2=hijk,e^2=k,f^2=gjk,b^a=bdfgjk,c^a=ce,c^b=cghi,d^b=dg,d^c=dik,e^a=ei,e^b=egh,e^c=eik,e^d=ek,f^a=fj,f^b=fijk,f^c=fghijk,f^d=fk,f^e=fik,g^a=gk,g^b=gk,g^c=gjk,g^e=gk,h^a=hi,h^b=hk,h^c=hk,h^d=hk,h^e=hk,h^f=hj,i^a=ik,i^b=ij,i^c=ik,i^f=ik,j^a=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341003
maximal normal subgroup of#72341003

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