Finite Group Databasebeta

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#72284909

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237593
commutator#2308C22×C4
Frattini, cosocle#20585C23
prime radical#72284481C2
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432954 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3304 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3304 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,13,17,28)(4,9)(5,10)(6,11)(7,12)(8,15,27,46)(14,41,44,39)(16,40,48,57)(18,29)(19,30)(20,31)(21,33)(22,34)(23,36)(24,38)(25,51,35,42)(26,47,56,45)(32,50)(37,53)(43,61,63,60)(49,58)(52,59)(54,62,55,64);(3,14,17,44)(8,25,27,35)(13,39,28,41)(15,42,46,51)(16,43,48,63)(21,53)(26,55,56,54)(33,37)(40,60,57,61)(45,64,47,62)(49,52)(58,59);(1,3,5,16)(2,8,10,26,12,27,34,56)(4,15,18,45,20,46,50,47)(6,14,21,43,24,44,53,63)(7,17,22,48)(9,28,29,57)(11,35,37,55)(13,32,40,31)(19,51,52,64)(23,42,49,62)(25,33,54,38)(30,41,58,61,36,39,59,60)

Polycyclic presentation:

2^11|a^2=jk,b^2=jk,c^2=e,d^2=hk,e^2=hi,f^2=hj,b^a=bjk,c^a=cd,c^b=cf,d^a=dh,d^c=dg,e^a=eghj,e^b=eg,e^d=ei,f^a=fjk,f^b=fhjk,f^c=fgh,f^d=fj,f^e=fijk,g^c=gi,g^d=gjk,g^e=gjk,g^f=gj,h^c=hjk,h^e=hk,i^a=ij,i^b=ijk,i^c=ijk,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286327
maximal normal subgroup of#72286327

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