Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of D3×H(3,𝔽3) |
| automorphism | Aut(G) = D3×#4208 (order 972) · Inn(G) = C32×D3 · Out(G) = C3×D3 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C1 |
| direct product | D3×C27⋊C3 |
| unambiguous semidirect product | C3⋊(C2×C27⋊C3) |
| unambiguous semidirect product | (D3×C27)⋊C3 |
| Fitting, Sylow | C3×C27⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C32×D3 |
| commutator, socle | C32 | C2×C3×C9 |
| Fitting | C3×C27⋊C3 | C2 |
| prime radical | C32×D3 | C9 |
| cosocle | C3×C9 | C2×C32 |
| hypercenter | C27⋊C3 | D3 |
| hypocenter | C3 | C2×C27⋊C3 |
| LCS | G ▷ C32 ▷ C3 | |
| UCS | C1 ◁ C9 ◁ C27⋊C3 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 30 points:
(2,3)(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30);(1,2,3)(4,10,24,13,27,17,28,5,16,7,20,29,22,26,8,12,11,23,14,18,30,6,19,15,21,9,25)99 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D27⋊C3, C2×D3×C27⋊C3, D9×C27⋊C3, … |
| maximal normal subgroup of | D3×D27⋊C3, C2×D3×C27⋊C3, D3×#6869, … |
| maximal quotient of | Dic3×C27⋊C3, C2×D3×C27⋊C3, D3×#6864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 114);