Finite Group DatabasebetaFinite Group Databasebeta| order | 450 |
| isoclinism | isoclinic class of C3⋊D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C3 |
| direct product | C5×C3⋊D15 |
| unambiguous semidirect product | C3⋊(C5×D15) |
| unambiguous semidirect product | C5⋊(C5×C3⋊D3) |
| Fitting, socle, Hall | C32×C52 ⋊ C2 |
| derived | C32×C5 ⋊ C2×C5 |
| Sylow | C52 ⋊ C3⋊D3 |
| Sylow | C32 ⋊ C5×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊D15 |
| commutator, cosocle, hypocenter | C32×C5 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C52 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C5 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3)(5,6)(7,8,9,10,11)(13,14)(15,16);(4,5,6)(12,13,15,16,14);(1,2,3)120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#4380, #11744709, C5×D5×C3⋊D3, … |
| maximal normal subgroup of | C5×#4380, #11744709, C5×D5×C3⋊D3, … |
| maximal quotient of | C5×C3⋊Dic15, C2×C5×C3⋊D15, C5×C3⋊D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(450, 32);