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#72286086

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72284401C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1902 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1902 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:

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

Polycyclic presentation:

2^11|b^2=jk,c^2=e,e^2=h,f^2=hijk,b^a=bjk,c^a=cd,c^b=cf,d^b=dg,d^c=dgj,e^a=egj,e^b=egj,e^d=eik,f^a=fgi,f^b=fhij,f^c=fghij,f^d=fik,f^e=fijk,g^c=gi,g^e=gj,g^f=gjk,h^a=hj,h^b=hj,h^d=hk,h^f=hk,i^b=ij,i^c=ij,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286190
maximal normal subgroup of#72286190

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