Finite Group Databasebeta

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#72241379

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#8934620 ⋊ D3
derived#72227582 ⋊ C2

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, cosocle, hypocenter#72227582C2
Frattini, hypercenterC23#87588
Fitting#8934620D3
LCSG ▷ #72227582
UCSC1 ◁ C22 ◁ C23

2749 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 2749 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2,3,2^9|c^2=j,d^2=j,e^2=k,f^2=k,b^a=b^2,c^a=cefijk,c^b=d,d^a=cdeik,d^b=cdj,d^c=dj,e^a=fk,e^b=ef,e^c=ei,e^d=ei,f^a=ek,f^b=e,f^c=fi,f^e=fk,g^a=h,g^b=h,g^c=gj,g^e=gk,h^a=g,h^b=gh,h^c=hj,h^d=hj,h^f=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72241381, #72241407, #72241410
maximal normal subgroup of#72241381, #72241407, #72241410

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