Finite Group DatabasebetaFinite Group Databasebeta| order | 150 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×GL(2,𝔽5) (order 2880) · Inn(G) = D3 · Out(G) = GL(2,𝔽5) |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C5 |
| direct product | C52×D3 |
| unambiguous semidirect product | C3⋊(C2×C52) |
| Fitting, socle, Hall | C3×C52 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C52 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C52 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C52 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 75 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3)(9,10,11,12,13);(1,2,3)(4,5,6,7,8)75 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×D5, D3×C5⋊D5, C2×C52×D3, … |
| maximal normal subgroup of | C5×D3×D5, D3×C5⋊D5, C2×C52×D3, … |
| maximal quotient of | C52×Dic3, C2×C52×D3, C52×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(150, 10);