Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of D3×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | D3×C9⋊C9 |
| unambiguous semidirect product | C3⋊(C2×C9⋊C9) |
| unambiguous semidirect product | (D3×C9)⋊C9 |
| Fitting, Sylow | C3×C9⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C32×D3 |
| commutator | C32 | C2×C3×C9 |
| Fitting | C3×C9⋊C9 | C2 |
| socle, cosocle | C33 | C2×C32 |
| prime radical | C32×D3 | C32 |
| hypercenter | C9⋊C9 | D3 |
| hypocenter | C3 | C2×C9⋊C9 |
| LCS | G ▷ C32 ▷ C3 | |
| UCS | C1 ◁ C32 ◁ C9⋊C9 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3)(13,16,21,15,19,17,18,14,20);(1,2,3)(4,8,12,6,11,7,9,5,10)(14,16,19)(17,21,20)99 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D9⋊C9, C2×D3×C9⋊C9, D9×C9⋊C9, … |
| maximal normal subgroup of | D3×D9⋊C9, C2×D3×C9⋊C9, D3×#6875, … |
| maximal quotient of | Dic3×C9⋊C9, C2×D3×C9⋊C9, D3×#6855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 97);