Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×C4×D3 (order 144) · Inn(G) = D3 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×D3×C9 |
| unambiguous semidirect product | C3⋊(C2×C5×C9) |
| Fitting, Hall | C3×C5×C9 ⋊ C2 |
| derived | C3 ⋊ C2×C5×C9 |
| Sylow | C3×C9 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C9 | D3 |
| commutator, hypocenter | C3 | C2×C5×C9 |
| Frattini | C3 | C3×C5×D3 |
| Fitting | C3×C5×C9 | C2 |
| socle | C32×C5 | C2×C3 |
| prime radical | C3×C5×D3 | C3 |
| cosocle | C32 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C5×C9 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3)(4,5,6,7,8)(9,10,12,11,13,15,14,16,17);(1,2,3)135 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C9×D5, C5×D3×D9, D3×D45, … |
| maximal normal subgroup of | D3×C9×D5, C5×D3×D9, D3×D45, … |
| maximal quotient of | C5×C9×Dic3, C2×C5×D3×C9, C5×C9×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 9);