Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C2×C3×D3×AGL(1,𝔽5) (order 720) · Inn(G) = D15 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×D15 |
| unambiguous semidirect product | C3⋊(C9×D5) |
| unambiguous semidirect product | C5⋊(D3×C9) |
| Fitting, Hall | C3×C5×C9 ⋊ C2 |
| derived | C3×C5 ⋊ C2×C9 |
| Sylow | C3×C9 ⋊ D5 |
| Sylow | C5 ⋊ D3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D15 |
| commutator, hypocenter | C3×C5 | C2×C9 |
| Frattini | C3 | C3×D15 |
| Fitting | C3×C5×C9 | C2 |
| socle, cosocle | C32×C5 | C2×C3 |
| prime radical | C3×D15 | C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ 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)(5,6)(7,8)(9,10,12,11,13,15,14,16,17);(1,2,3)(4,5,7,8,6)81 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C9×D5, D9×D15, D15⋊D9, … |
| maximal normal subgroup of | D3×C9×D5, D9×D15, D15⋊D9, … |
| maximal quotient of | C9×Dic15, C2×C9×D15, C9×D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 13);