Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| isoclinism | isoclinic class of C5⋊AGL(1,ℤ/9) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,ℤ/9) (order 1080) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊AGL(1,ℤ/9) |
| unambiguous semidirect product | 3−1+2⋊D5 |
| unambiguous semidirect product | D45⋊C3 |
| Fitting, Hall | C5×3−1+2 ⋊ C2 |
| derived | C5×C9 ⋊ C2×C3 |
| Sylow | 3−1+2 ⋊ D5 |
| Sylow | C5 ⋊ AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×C9 | C2×C3 |
| Frattini | C3 | C3×D15 |
| Fitting | C5×3−1+2 | C2 |
| socle | C3×C5 | C3×D3 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5)(7,11,14,9,10,13)(8,12);(1,2,4,5,3)(6,14,9,12,10,11,8,7,13)Polycyclic presentation:
2,3^2,5,3|c^3=e^2,c^a=c^2e,d^a=d^4,e^a=e^2,c^b=ce32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 15);