Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊D53) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D159 |
| unambiguous semidirect product | C53⋊(C3⋊AGL(1,𝔽7)) |
| unambiguous semidirect product | D1113⋊C3 |
| Fitting, socle, derived | C3×C7×C53 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8)×C53 ⋊ C2 |
| Hall | C7×C53 ⋊ C3×D3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ D53 |
| Sylow | C53 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3×D159 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C7×C53 | C2×C3 |
| LCS | G ▷ C3×C7×C53 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,53|c^a=c^2,d^a=d^6,e^a=e^52,d^b=d^2402 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 18);