Finite Group DatabasebetaFinite Group Databasebeta| order | 150 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C4×C5×D3 (order 120) · Inn(G) = D3 · Out(G) = C4×C5 |
| holomorph | G ⋊ Aut(G) = D32×AGL(1,ℤ/25) |
| homology | Abelianization (H1) = C2×C25 · Schur multiplier (H2) = C1 |
| direct product | D3×C25 |
| unambiguous semidirect product | C3⋊(C2×C25) |
| Fitting, Hall | C3×C25 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | D3 |
| commutator, hypocenter | C3 | C2×C25 |
| Frattini | C5 | C5×D3 |
| Fitting | C3×C25 | C2 |
| socle, cosocle | C3×C5 | C2×C5 |
| prime radical | C5×D3 | C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C25 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 75 |
A minimal faithful permutation representation — 2 generators on 28 points:
(2,3)(4,5,7,10,14,6,8,11,15,19,9,12,16,20,23,13,17,21,24,26,18,22,25,27,28);(1,2,3)75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D25, C2×D3×C25, D9×C25, … |
| maximal normal subgroup of | D3×D25, C2×D3×C25, C3×D3×C25, … |
| maximal quotient of | Dic3×C25, C2×D3×C25, D9×C25, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(150, 1);