Finite Group Databasebeta

← Prev (#72310922) Next (#72310924) →

#72310923

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#35346D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2080 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2080 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:

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

Polycyclic presentation:

2^11|b^2=jk,c^2=dfk,d^2=hk,f^2=k,b^a=bj,c^a=ce,c^b=cd,d^a=dg,d^b=dh,d^c=dh,e^b=egij,e^c=egijk,e^d=eijk,f^a=fij,f^b=fhk,f^c=fh,f^e=fk,g^b=gj,g^c=gi,g^d=gj,g^f=gk,h^a=hj,h^e=hk,i^b=ij,i^c=ijk,i^d=ik,j^c=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311013, #72311187
maximal normal subgroup of#72311013, #72311187

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