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#72284136

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#251828C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2018 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2018 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=jk,c^2=gh,d^2=h,e^2=hijk,f^2=hk,b^a=bf,c^a=ce,d^c=dg,e^a=eghijk,e^b=eghj,e^c=ehj,e^d=eijk,f^a=fhk,f^b=fh,f^c=fghi,f^e=fij,g^a=gi,g^c=gj,g^d=gj,g^e=gjk,g^f=gjk,h^c=hj,h^e=hk,i^a=ij,i^b=ij,i^c=ijk,i^d=ik,i^e=ik,i^f=ik,j^a=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285547, #72340464, #72340497
maximal normal subgroup of#72285547, #72340464, #72340497

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