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#72340934

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#72284399C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2125 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2125 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)(3,14,16,43)(4,19,21,54)(6,11,29,44)(7,13,26,42)(8,27,32,52)(9,36,37,50)(12,24,33,18)(15,48,49,61)(17,46,53,35)(20,38,51,64)(22,47,59,60)(23,55,58,31)(25,34,30,45)(28,39,57,62)(40,63,41,56);(1,3)(2,8,13,35)(4,20,23,15)(5,24,26,16)(6,27)(7,18)(9,31,40,54)(10,32)(11,43,45,33)(12,34)(14,44)(17,30)(19,37)(21,49)(22,50,28,56)(25,52,29,53)(36,57)(38,62,48,60)(39,61)(41,55)(42,46)(47,64)(51,58)(59,63);(1,4)(2,9,13,40)(3,15)(5,21,26,58)(6,28,30,22)(7,23)(8,31)(10,41)(11,38)(12,39)(14,47)(16,51,24,49)(17,50,27,56)(18,20)(19,46,55,32)(25,57)(29,59)(33,62,43,60)(34,61,44,64)(35,54)(36,53)(37,42)(45,48)(52,63)

Polycyclic presentation:

2^11|a^2=d,b^2=ghijk,c^2=hjk,f^2=gijk,b^a=bde,c^a=cefgijk,c^b=chi,d^b=dik,d^c=dg,e^a=ei,e^b=ek,e^c=egh,e^d=ek,f^a=fijk,f^b=fhj,f^c=fhijk,f^e=fi,g^a=gk,g^b=gjk,g^c=gk,g^e=gk,g^f=gk,h^a=hi,h^b=hj,h^d=hk,h^e=hk,h^f=hj,i^a=ik,i^b=ik,i^c=ij,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.9*F.10*F.11);;
gap> SetPower(coll, 3, F.8*F.10*F.11);;
gap> SetPower(coll, 6, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
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.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
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);