Finite Group Databasebeta

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#72268230

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268218 ⋊ C3
derived#8093108 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#35346⋊C3
commutator, cosocle, hypocenter#8093108C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72268218C3
hypercenterC23#85645
LCSG ▷ #8093108
UCSC1 ◁ C22 ◁ C23

3910 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 3910 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 32

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

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

Polycyclic presentation:

2,3,2^9|c^2=i,d^2=i,e^2=j,f^2=j,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=di,e^b=ef,e^d=eij,f^b=e,f^c=fij,f^e=fj,g^b=h,g^d=gk,g^e=gk,h^b=ghk,h^c=hk,h^d=hk,h^f=hk,i^a=ik

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72268240

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