Finite Group DatabasebetaFinite Group Databasebeta| order | 450 |
| isoclinism | isoclinic class of C3⋊(C5⋊D5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| direct product | C3×C3⋊(C5⋊D5) |
| unambiguous semidirect product | C3⋊(C3×C5⋊D5) |
| unambiguous semidirect product | C5⋊(C3×D15) |
| Fitting, socle, Hall | C32×C52 ⋊ C2 |
| derived | C3×C52 ⋊ C2×C3 |
| Sylow | C52 ⋊ C3×D3 |
| Sylow | C32 ⋊ C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C5⋊D5) |
| commutator, cosocle, hypocenter | C3×C52 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C52 | C2 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C3 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 150 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3)(5,6)(8,9)(10,11)(13,14)(15,16);(4,5,6)(12,13,15,16,14);(7,8,10,11,9)117 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×C5⋊D5, C3×D5×D15, D3×C3⋊(C5⋊D5), … |
| maximal normal subgroup of | C3×D3×C5⋊D5, C3×D5×D15, D3×C3⋊(C5⋊D5), … |
| maximal quotient of | C3×C3⋊#1039, C2×C3×C3⋊(C5⋊D5), C3×C5⋊D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(450, 30);