Finite Group Databasebeta

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#72258569

order4608
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

FittingC22×#399 ⋊ C3⋊D3
derivedA4×#95⋊C3 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#72219738
commutator, cosocle, hypocenterA4×#95⋊C3C22
FrattiniC23C2×A4⋊S4
FittingC22×#399C3⋊D3
socleC25C2×C3⋊S4
LCSG ▷ A4×#95⋊C3
UCSC1 ◁ C2

1877 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1877 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 24
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,4)(5,13,15)(6,8,7,11,9,14)(10,16,12)(17,18,21,20,24,19)(22,23);(2,3)(6,7)(11,14)(12,16)(13,15)(18,19)(21,24)(22,23);(2,4,3)(5,6,7)(8,15,13)(9,12,16)(10,11,14)(18,21,22)(19,23,24)

Polycyclic presentation:

2^2,3^2,2^7|e^2=i,f^2=j,c^a=c^2,d^a=d^2,e^a=fjk,e^b=eik,e^c=fj,e^d=efi,f^a=eik,f^b=fjk,f^c=efjk,f^d=ei,f^e=fk,g^a=h,g^c=gh,g^d=gij,h^a=g,h^c=g,h^d=hjk,i^a=j,i^c=j,i^d=ijk,j^a=i,j^c=ijk,j^d=i

Conjugacy classes & character table

60 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, 3, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.6*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);