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#72341244

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#251827
commutator, Frattini, cosocleC22×#127C24
prime radical#7672751C22
LCSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1

5342 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 5342 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^11|b^2=jk,c^2=k,d^2=gj,e^2=j,f^2=jk,b^a=be,c^a=cf,c^b=cg,d^a=defhijk,d^b=dj,d^c=dj,e^a=ej,e^b=ej,e^c=eh,e^d=ehi,f^a=fjk,f^b=fhij,f^c=fjk,f^d=fh,g^a=gijk,g^e=gj,g^f=gjk,h^b=hj,h^d=hjk,i^b=ij,i^c=ijk,i^d=ik

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

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