Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D5×D21 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽5)×AGL(1,𝔽7) (order 10080) · Inn(G) = D5×D21 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D5×D21 |
| unambiguous semidirect product | (C3×D5)⋊(C3×D7) |
| Fitting, socle, Hall | C32×C5×C7 ⋊ C22 |
| derived | C3×C5×C7 ⋊ C22×C3 |
| Hall | C32×C7 ⋊ C2×D5 |
| Hall | C32×C5 ⋊ C2×D7 |
| Hall | C5×C7 ⋊ C2×C3×D3 |
| Sylow | C32 ⋊ D5×D7 |
| Sylow | C7 ⋊ C3×D3×D5 |
| Sylow | C5 ⋊ C2×C3×D21 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D5×D21 |
| commutator, cosocle, hypocenter | C3×C5×C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7 | C22 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C3 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 210 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D5×C11×D21, C3×D5×D231, C3×D21×D55, … |
| maximal normal subgroup of | C3×D5×C11×D21, C3×D5×C13×D21, C3×D5×C17×D21, … |
| maximal quotient of | C3×D5×C11×D21, C3×D5×D231, C3×D21×D55, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 98);