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#72255291

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

Aliases

No aliases for this group.

Splittings

FittingC3×#23907 ⋊ C2
SylowC3 ⋊ #144388

Subgroups

subgroupG/H
centerC22#71671
commutator, cosocleC23×C3×D4C23
FrattiniC23×D4C22×D3
FittingC3×#23907C2
socleC22×C3#2021
prime radical#10826768C2
hypercenter#23907D3
hypocenterC3#144388
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #23907

2844 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 2844 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) = 27
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

(1,3,4,8)(2,6)(5,7)(9,11)(10,16,19,15)(12,22,13,21)(14,23)(17,24)(18,20)(26,27);(1,3,4,8)(2,5,7,6)(9,11)(10,15)(12,22)(13,21)(14,23)(16,19)(17,20)(18,24)(25,26,27);(1,2,4,7)(3,5,8,6)(9,10)(11,20,23,24)(12,17)(13,18)(14,19)(15,22,16,21)

Polycyclic presentation:

2^9,3|a^2=h,b^2=gi,c^2=dfgi,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#72270439, D3×#144388, #72270536
maximal normal subgroup of#72270439, D3×#144388, #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.8);;
gap> SetPower(coll, 2, F.7*F.9);;
gap> SetPower(coll, 3, F.4*F.6*F.7*F.9);;
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);