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#72341241

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

5332 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 5332 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:

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

Polycyclic presentation:

2^11|a^2=ij,b^2=jk,c^2=k,e^2=j,f^2=jk,b^a=be,c^a=cf,c^b=cg,d^a=dk,d^b=dgjk,d^c=dg,e^b=ej,e^c=eh,e^d=ei,f^b=fhijk,f^c=fjk,f^d=fik,g^a=gij,g^e=gj,g^f=gjk,h^b=hj,h^d=hk,i^b=ij,i^c=ijk

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