Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D3×D21 |
| automorphism | Aut(G) = C4×D32×AGL(1,𝔽7) (order 6048) · Inn(G) = D3×D21 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D3×D21 |
| unambiguous semidirect product | (C3×D3)⋊(C5×D7) |
| unambiguous semidirect product | (D3×C7)⋊(C5×D3) |
| Fitting, socle, Hall | C32×C5×C7 ⋊ C22 |
| derived, Hall | C32×C7 ⋊ C22×C5 |
| Hall | C32×C5 ⋊ C2×D7 |
| Hall | C5×C7 ⋊ D32 |
| Sylow | C32 ⋊ C2×C5×D7 |
| Sylow | C7 ⋊ C5×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D3×D21 |
| commutator, cosocle, hypocenter | C32×C7 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7 | C22 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C5 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 210 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 108);