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#72252958

order5184
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2#72220152
commutator, cosocleC3×#41⋊C9C22×C3
FrattiniC3×C4C2×C3×C3⋊S4
FittingC33×H(5,𝔽2)D3
socleC2×C32C3×GL(2,ℤ/4)
hypercenterQ8C3⋊(((C22⋊C9)⋊C2)⋊C3)
hypocenterC3×Q8⋊C9C3×D4
LCSG ▷ C3×#41⋊C9 ▷ C3×Q8⋊C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

1155 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 36
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 3 generators on 36 points:

(1,9)(2,5)(3,7)(4,14)(6,10)(8,13)(12,16)(17,21)(20,24)(22,23)(25,26,27)(29,32,36)(31,35,33);(2,5)(3,9)(4,12)(6,10)(8,16)(11,20)(13,23)(14,22)(15,17)(18,24)(19,21)(26,27)(29,31)(30,34)(32,35)(33,36);(1,4,12)(2,8,17)(3,11,21)(5,15,16)(6,13,24)(7,14,22)(9,19,20)(10,18,23)(28,36,35,30,29,31,34,32,33)

Polycyclic presentation:

2^2,3,2,3^3,2^3|b^2=j,d^2=j,e^3=g,h^2=j,i^2=j,b^a=bd,d^a=dj,d^b=dj,e^a=e^2g^2,e^c=eg,f^a=f^2,g^a=g^2,h^a=ij,h^e=hi,i^a=hj,i^e=h,i^h=ij

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252975, C2×#72252958
maximal normal subgroup ofC2×#72252958
maximal quotient ofC2×#72252958

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