Finite Group DatabasebetaFinite Group Databasebeta| order | 300 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×C5×D3 (order 240) · Inn(G) = D3 · Out(G) = C2×C4×C5 |
| homology | Abelianization (H1) = C22×C25 · Schur multiplier (H2) = C2 |
| direct product | C2×D3×C25 |
| unambiguous semidirect product | C3⋊(C22×C25) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C25) |
| Fitting | C2×C3×C25 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C25 |
| Hall | C3×C25 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C25 | D3 |
| commutator, hypocenter | C3 | C22×C25 |
| Frattini | C5 | C2×C5×D3 |
| Fitting | C2×C3×C25 | C2 |
| socle | C2×C3×C5 | C2×C5 |
| prime radical | C2×C5×D3 | C5 |
| cosocle | C3×C5 | C22×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C25 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 150 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2)(4,5)(6,10,18,26,30,8,14,23,29,16,11,19,27,12,21,15,24,9,17,25,20,7,13,22,28);(1,2)(3,4,5)150 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic25, #10597031, #10597032, … |
| maximal normal subgroup of | D3×Dic25, #10597031, #10597032, … |
| maximal quotient of | Dic6×C25, C4×D3×C25, D12×C25, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(300, 10);