Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C53 · Schur multiplier (H2) = C3 |
| direct product | C7×C3⋊D3×C53 |
| unambiguous semidirect product | C3⋊(D3×C7×C53) |
| Fitting, socle, Hall | C32×C7×C53 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C7×C53 |
| Hall | C32×C53 ⋊ C2×C7 |
| Hall | C32×C7 ⋊ C2×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C53 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C7×C53 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C53 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C7×C53 | |
48 conjugacy classes of subgroups.
2226 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×C3⋊D3×D53, D7×C3⋊D3×C53, C3⋊D3×D371, … |
| maximal normal subgroup of | C7×C3⋊D3×D53, D7×C3⋊D3×C53, C3⋊D3×D371, … |
| maximal quotient of | C7×C3⋊Dic3×C53, C2×C7×C3⋊D3×C53, C5×C7×C3⋊D3×C53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 29);