Finite Group Databasebeta

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#72241402

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting#13378 ⋊ D3
derived#73291 ⋊ C22

Subgroups

subgroupG/H
center, socleC22#73026
commutator, cosocle, hypocenter#73291C22
FrattiniC22×Q8C2×S4
Fitting#13378D3
hypercenterC23#1020⋊C2
LCSG ▷ #73291
UCSC1 ◁ C22 ◁ C23

715 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 20 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 — 2 generators on 24 points:

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

Polycyclic presentation:

2^2,3,2^7|d^2=i,e^2=i,f^2=j,g^2=j,c^a=c^2,c^b=c^2,d^a=di,d^b=dfghij,d^c=e,e^a=de,e^b=defhj,e^c=dei,e^d=ei,f^a=gj,f^b=gj,f^c=fg,f^d=fh,f^e=fh,g^a=fj,g^b=fj,g^c=f,g^d=gh,g^f=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72241407

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