Finite Group DatabasebetaFinite Group Databasebeta| order | 450 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×C5×D3 (order 240) · Inn(G) = D3 · Out(G) = C2×C4×C5 |
| homology | Abelianization (H1) = C2×C3×C25 · Schur multiplier (H2) = C1 |
| direct product | C3×D3×C25 |
| unambiguous semidirect product | C3⋊(C2×C3×C25) |
| Fitting, Hall | C32×C25 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C25 |
| Sylow | C32 ⋊ C2×C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C25 | D3 |
| commutator, hypocenter | C3 | C2×C3×C25 |
| Frattini | C5 | C3×C5×D3 |
| Fitting | C32×C25 | C2 |
| socle | C32×C5 | C2×C5 |
| prime radical | C3×C5×D3 | C5 |
| cosocle | C3×C5 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C25 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 150 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3)(5,6)(7,11,19,27,31,9,15,24,30,17,12,20,28,13,22,16,25,10,18,26,21,8,14,23,29);(4,5,6)225 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D25, D32×C25, D3×D75, … |
| maximal normal subgroup of | C3×D3×D25, D32×C25, D3×D75, … |
| maximal quotient of | C3×Dic3×C25, C2×C3×D3×C25, C3×D9×C25, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(450, 6);