Finite Group DatabasebetaFinite Group Databasebeta| order | 1350 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C25 · Schur multiplier (H2) = C1 |
| direct product | C25×AGL(1,ℤ/9) |
| unambiguous semidirect product | D9⋊(C3×C25) |
| unambiguous semidirect product | 3−1+2⋊(C2×C25) |
| unambiguous semidirect product | (D9×C25)⋊C3 |
| Fitting, Hall | C25×3−1+2 ⋊ C2 |
| derived | C9 ⋊ C2×C3×C25 |
| Sylow | 3−1+2 ⋊ C2×C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C2×C3×C25 |
| Frattini, socle | C3×C5 | C3×C5×D3 |
| Fitting | C25×3−1+2 | C2 |
| prime radical | C5×AGL(1,ℤ/9) | C5 |
| cosocle | C5×C9 | C2×C3×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C25 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 225 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(27,31,34,29,30,33)(28,32);(26,34,29,32,30,31,28,27,33)250 conjugacy classes — view the full character table.
| maximal subgroup of | D25×AGL(1,ℤ/9), AGL(1,ℤ/9)×5−1+2 |
| maximal normal subgroup of | D25×AGL(1,ℤ/9), AGL(1,ℤ/9)×5−1+2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1350, 11);