Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×#4208 (order 1944) · Inn(G) = D3 · Out(G) = C2×#4208 |
| homology | Abelianization (H1) = C2×C3×C27 · Schur multiplier (H2) = C3 |
| direct product | C3×D3×C27 |
| unambiguous semidirect product | C3⋊(C2×C3×C27) |
| Fitting, Sylow | C32×C27 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C27 | D3 |
| commutator, hypocenter | C3 | C2×C3×C27 |
| Frattini | C9 | C32×D3 |
| Fitting | C32×C27 | C2 |
| socle | C33 | C2×C9 |
| prime radical | C32×D3 | C9 |
| cosocle | C3×C9 | C2×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C27 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 33 points:
(5,6)(7,8,11,9,12,17,14,19,24,10,13,18,15,20,25,22,27,30,16,21,26,23,28,31,29,32,33);(1,2,3)(4,5,6)243 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D27, D32×C27, D3×C3⋊D27, … |
| maximal normal subgroup of | C3×D3×D27, D32×C27, D3×C3⋊D27, … |
| maximal quotient of | C3×Dic3×C27, C2×C3×D3×C27, D3×C9×C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 112);