Finite Group DatabasebetaFinite Group Databasebeta| order | 150 |
| isoclinism | isoclinic class of D15 |
| automorphism | Aut(G) = C4×D3×AGL(1,𝔽5) (order 480) · Inn(G) = D15 · Out(G) = C42 |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D15 |
| unambiguous semidirect product | C3⋊(C5×D5) |
| unambiguous semidirect product | C5⋊(C5×D3) |
| Fitting, socle, Hall | C3×C52 ⋊ C2 |
| derived | C3×C5 ⋊ C2×C5 |
| Sylow | C52 ⋊ D3 |
| Sylow | C3 ⋊ C5×D5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C5 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3)(4,5,6,7,8)(10,11)(12,13);(1,2,3)(9,10,12,13,11)45 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×D5, D5×D15, D15⋊D5, … |
| maximal normal subgroup of | C5×D3×D5, D5×D15, D15⋊D5, … |
| maximal quotient of | C5×Dic15, C2×C5×D15, C5×D45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(150, 11);