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#72255262

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×#24056C2
socleC22×C3#2021
prime radicalC3×#3484C22
hypercenter#24056D3
hypocenterC3#144415
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #24056

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:

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

Polycyclic presentation:

2^9,3|a^2=d,c^2=deh,b^a=bdf,c^a=cfg,c^b=cde,d^b=dg,d^c=dg,e^a=eh,e^b=eg,e^c=egi,f^a=fg,f^b=fg,f^c=fh,f^e=fi,h^b=hi,h^c=hi,j^a=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#144415, #72255289, #72270168
maximal normal subgroup ofD3×#144415, #72255289, #72270168

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