Finite Group Databasebeta

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#72270002

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

FittingC3×#7304 ⋊ C2
SylowC3 ⋊ #144386

Subgroups

subgroupG/H
centerC22#68965
commutator, cosocleC23×C3×D4C23
FrattiniC23×D4C22×D3
FittingC3×#7304C2
socleC22×C3#2021
hypercenter#7304D3
hypocenterC3#144386
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #7304

3576 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 3576 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) = 48

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

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=fgh,c^2=dfhi,e^2=i,b^a=bef,c^a=cfhi,c^b=cd,d^a=dh,d^b=dg,d^c=dg,e^b=eghi,e^c=egh,e^d=ei,f^a=fghi,f^b=fi,f^c=fg,f^e=fi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72270007, D3×#144386, #72270536
maximal normal subgroup of#72270007, D3×#144386, #72270536

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