Finite Group Databasebeta

← Prev (#72255269) Next (#72255271) →

#72255270

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

SylowC3 ⋊ #144414

Subgroups

subgroupG/H
centerC22#71671
commutator, cosocleC23×C3×D4C23
FrattiniC23×D4C22×D3
Fitting, prime radicalC3×#23906C2
socleC22×C3#2021
hypercenter#23906D3
hypocenterC3#144414
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #23906

2300 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 2300 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 19
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

2^9,3|a^2=hi,b^2=gi,c^2=df,d^2=i,e^2=i,b^a=be,c^a=cd,c^b=cdf,d^b=dg,d^c=dh,e^b=ei,e^c=eghi,f^a=fhi,f^b=fg,f^c=fh,f^d=fi,f^e=fi,h^b=hi,h^c=hi,j^c=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72255279, #72270439, D3×#144414
maximal normal subgroup of#72255279, #72270439, D3×#144414

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