Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C33 |
| direct product | C9×C3⋊(C3⋊D3) |
| unambiguous semidirect product | C3⋊(C9×C3⋊D3) |
| Fitting, Sylow | C33×C9 ⋊ C2 |
| derived | C33 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C3⋊(C3⋊D3) |
| commutator, hypocenter | C33 | C2×C9 |
| Frattini | C3 | C3×C3⋊(C3⋊D3) |
| Fitting | C33×C9 | C2 |
| socle, cosocle | C34 | C2×C3 |
| prime radical | C3×C3⋊(C3⋊D3) | C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C9 | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 18 points:
(2,3)(5,6)(8,9)(10,11,13,12,14,16,15,17,18);(7,8,9);(4,5,6);(1,2,3)135 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C9×C3⋊D3, D9×C3⋊(C3⋊D3), #11758088, … |
| maximal normal subgroup of | D3×C9×C3⋊D3, D9×C3⋊(C3⋊D3), #11758088, … |
| maximal quotient of | C9×C3⋊(C3⋊Dic3), C2×C9×C3⋊(C3⋊D3), C9×C3⋊(C3⋊D9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 241);