Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D105 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D105 |
| unambiguous semidirect product | (C2×C3)⋊(C3×D35) |
| unambiguous semidirect product | (C2×C5)⋊(C3×D21) |
| Fitting, socle | C2×C32×C5×C7 ⋊ C2 |
| derived | C3×C5×C7 ⋊ C22×C3 |
| Hall | C32×C5×C7 ⋊ C22 |
| Hall | C32×C7 ⋊ C2×D5 |
| Hall | C32×C5 ⋊ C2×D7 |
| Hall | C5×C7 ⋊ C2×C3×D3 |
| Sylow | C32 ⋊ C2×D35 |
| Sylow | C7 ⋊ C2×C3×D15 |
| Sylow | C5 ⋊ C2×C3×D21 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D105 |
| commutator, cosocle, hypocenter | C3×C5×C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5×C7 | C2 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2×C3 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D11×D105, C2×C3×D1155, C2×C3×C11×D105, … |
| maximal normal subgroup of | C2×C3×C11×D105, C2×C3×C13×D105, C2×C3×C17×D105, … |
| maximal quotient of | C3×D11×D105, C2×C3×D1155, C2×C3×C11×D105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 126);