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#72237331

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#93190
commutatorC22×#117C42
Frattini, cosocle#607575C22
prime radicalC25#342
LCSG ▷ C22×#117 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4 ◁ #607575 ◁ G
LECSG ▷ #607575 ▷ C24×C4 ▷ C23 ▷ C2 ▷ C1

1204 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 1204 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,8,6,10,5,9)(3,11,12,15,14,16,13,7)(17,18,20,24)(19,28,29,46,33,41,30,43)(21,36,38,45)(22,27,34,44)(23,32,40,31,42,37,48,47)(25,39,26,35);(1,3,4,13,6,14,5,12)(2,7,9,11,10,15,8,16)(17,19,20,30,22,33,34,29)(18,23,25,41,27,42,45,28)(21,37,38,31,39,32,35,47)(24,40,26,46,44,48,36,43)

Polycyclic presentation:

2^11|a^2=c,b^2=d,c^2=h,d^2=hijk,e^2=hik,g^2=k,b^a=be,c^b=cf,d^a=dg,d^c=di,e^a=efhik,e^b=eghi,e^c=ejk,e^d=eijk,f^a=fj,f^b=fi,f^c=fk,f^d=fk,f^e=fk,g^a=gik,g^b=gij,g^c=gk,g^d=gk,g^e=gk,h^b=hk,i^a=ik,i^b=ik,j^a=jk

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72237332

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