Finite Group Databasebeta

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#72311205

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#20586C23
prime radical#72310879C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2071 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2071 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)(3,13,18,40)(4,19,22,10)(5,11,24,28)(6,12,26,29)(7,8)(9,16)(14,38,44,37)(15,31)(17,39)(20,34)(21,35)(23,50)(25,36)(27,48)(30,42)(32,47)(33,59,54,46)(41,57,51,55)(43,60)(45,53)(49,62)(52,61)(56,63,58,64);(2,8)(3,14)(9,30)(10,19)(11,28)(12,29)(13,37)(15,45)(16,47)(17,43)(18,44)(20,23)(27,32)(33,56)(35,36)(38,40)(39,61)(42,48)(46,63)(49,52)(54,58)(55,57)(59,64)(60,62);(1,3,15,46)(2,9,31,39)(4,14,41,63)(5,16,6,17)(7,18,45,59)(8,27,53,62)(10,30,55,60)(11,13,12,33)(19,32,57,61)(20,42,21,43)(22,44,51,64)(23,47,25,52)(24,48,26,49)(28,40,29,54)(34,37,35,56)(36,58,50,38)

Polycyclic presentation:

2^11|a^2=ghij,c^2=ef,d^2=h,b^a=bk,c^a=ce,c^b=cd,d^a=dg,d^b=dh,d^c=dgi,e^a=eijk,e^b=egij,e^d=eij,f^a=fijk,f^b=fh,f^d=fjk,g^a=gjk,g^b=gj,g^c=gi,g^d=gj,g^f=gk,h^a=hj,h^c=hjk,h^e=hk,i^a=ik,i^b=ij,i^c=ik,i^d=ik,j^c=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311339, #72340325, #72340466
maximal normal subgroup of#72311339, #72340325, #72340466

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