Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C53 · Schur multiplier (H2) = C3 |
| direct product | C32×D7×C53 |
| Fitting, socle, Hall | C32×C7×C53 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C32×C53 |
| Hall | C7×C53 ⋊ C2×C32 |
| Hall | C32×C7 ⋊ C2×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C53 | D7 |
| commutator, hypocenter | C7 | C2×C32×C53 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C53 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32×C53 | |
48 conjugacy classes of subgroups.
2385 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D7×D53, C3×D3×D7×C53, C3×D7×D159, … |
| maximal normal subgroup of | C32×D7×D53, C3×D3×D7×C53, C3×D7×D159, … |
| maximal quotient of | C32×Dic7×C53, C2×C32×D7×C53, C32×C5×D7×C53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 23);