Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D3×D15 |
| automorphism | Aut(G) = C2×C3×D32×AGL(1,𝔽5) (order 4320) · Inn(G) = D3×D15 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | D3×C7×D15 |
| unambiguous semidirect product | (C3×D3)⋊(C7×D5) |
| unambiguous semidirect product | (C5×D3)⋊(D3×C7) |
| Fitting, socle, Hall | C32×C5×C7 ⋊ C22 |
| derived, Hall | C32×C5 ⋊ C22×C7 |
| Hall | C32×C7 ⋊ C2×D5 |
| Hall | C5×C7 ⋊ D32 |
| Sylow | C32 ⋊ C2×C7×D5 |
| Sylow | C5 ⋊ D32×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D3×D15 |
| commutator, cosocle, hypocenter | C32×C5 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C7 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 210 |
189 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×C11×D15, D3×C7×D165, C7×D33⋊D15, … |
| maximal normal subgroup of | D3×C7×C11×D15, D3×C7×C13×D15, D3×C7×C17×D15, … |
| maximal quotient of | D3×C7×C11×D15, D3×C7×D165, C7×D33⋊D15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 110);