Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| automorphism | Aut(G) order 6250608 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D3339 |
| unambiguous semidirect product | C7⋊D477 |
| unambiguous semidirect product | C9⋊D371 |
| unambiguous semidirect product | C53⋊D63 |
| Fitting, derived, Hall | C7×C9×C53 ⋊ C2 |
| Hall | C9×C53 ⋊ D7 |
| Hall | C7×C53 ⋊ D9 |
| Hall | C7×C9 ⋊ D53 |
| Sylow | C53 ⋊ D63 |
| Sylow | C9 ⋊ D371 |
| Sylow | C7 ⋊ D477 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C7×C9×C53 | C2 |
| Frattini | C3 | D1113 |
| socle | C3×C7×C53 | D3 |
| LCS | G ▷ C7×C9×C53 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7,53,3|b^3=e^2,b^a=b^2e,c^a=c^6,d^a=d^52,e^a=e^21671 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×D63, D7×D477, D9×D371, … |
| maximal normal subgroup of | D53×D63, D7×D477, D9×D371, … |
| maximal quotient of | Dic3339, C2×D3339, C5×D3339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 9);