Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| isoclinism | isoclinic class of C3⋊D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C53 · Schur multiplier (H2) = C3 |
| direct product | C53×C3⋊D21 |
| unambiguous semidirect product | C3⋊(D21×C53) |
| unambiguous semidirect product | C7⋊(C3⋊D3×C53) |
| Fitting, socle, Hall | C32×C7×C53 ⋊ C2 |
| derived, Hall | C32×C7 ⋊ C2×C53 |
| Hall | C32×C53 ⋊ D7 |
| Hall | C7×C53 ⋊ C3⋊D3 |
| Sylow | C32 ⋊ D7×C53 |
| Sylow | C7 ⋊ C3⋊D3×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C53 | C3⋊D21 |
| commutator, hypocenter | C32×C7 | C2×C53 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C53 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C53 | |
48 conjugacy classes of subgroups.
1749 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×C3⋊D21, D7×C3⋊D3×C53, D371⋊(C3⋊D3), … |
| maximal normal subgroup of | D53×C3⋊D21, D7×C3⋊D3×C53, D371⋊(C3⋊D3), … |
| maximal quotient of | C53×C3⋊Dic21, C2×C53×C3⋊D21, C5×C53×C3⋊D21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 31);