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#72255264

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

Aliases

No aliases for this group.

Splittings

SylowC3 ⋊ #144415

Subgroups

subgroupG/H
centerC22#71671
commutator, cosocleC23×C3×D4C23
FrattiniC23×D4C22×D3
FittingC3×#23907C2
socleC22×C3#2021
prime radicalC3×#3484C22
hypercenter#23907D3
hypocenterC3#144415
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #23907

1884 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1884 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)(2,5)(4,8)(6,7)(9,12,14,19)(10,16)(11,13)(15,17,21,24)(18,23)(20,22)(26,27);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21)(25,26,27);(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,22,24)(11,16,19,15,20,23,12,21)

Polycyclic presentation:

2^9,3|a^2=e,c^2=dh,e^2=i,b^a=beh,c^a=cf,c^b=cd,d^a=dh,d^c=di,e^c=eg,e^d=ei,f^a=fg,f^b=fgh,f^c=fhi,f^d=fi,f^e=fi,g^a=gi,g^b=gi,g^c=gi,h^a=hi,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 ofD3×#144415, #72270439, #72270525
maximal normal subgroup ofD3×#144415, #72270439, #72270525

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, 3, F.4*F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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);