Finite Group DatabasebetaFinite Group Databasebeta| order | 780 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C13 · Schur multiplier (H2) = C2 |
| direct product | D3×D5×C13 |
| unambiguous semidirect product | (C5×D3)⋊(C2×C13) |
| unambiguous semidirect product | (C3×D5)⋊(C2×C13) |
| unambiguous semidirect product | D15⋊(C2×C13) |
| Fitting, socle, Hall | C3×C5×C13 ⋊ C22 |
| derived, Hall | C3×C5 ⋊ C22×C13 |
| Hall | C5×C13 ⋊ C2×D3 |
| Hall | C3×C13 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×D3×C13 |
| Sylow | C3 ⋊ C2×D5×C13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C13 | D3×D5 |
| commutator, cosocle, hypocenter | C3×C5 | C22×C13 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C13 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C13 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 195 |
156 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C13×AGL(1,𝔽5), D3×#63120, D3×D5×D13, … |
| maximal normal subgroup of | D3×C13×AGL(1,𝔽5), D3×#63120, D3×D5×D13, … |
| maximal quotient of | D3×C13×Dic5, D5×Dic3×C13, C13×#1200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(780, 39);