Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C53 · Schur multiplier (H2) = C3 |
| Fitting, socle | C3×C7×C53 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C2×C32×C53 |
| Hall | C3×ΓL(1,𝔽8)×C53 ⋊ C2 |
| Hall | D7×C53 ⋊ C32 |
| Hall | C7×C53 ⋊ C2×C32 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×C53 |
| Hall | D7 ⋊ C32×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C53 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C32×C53 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C53 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C53 | |
48 conjugacy classes of subgroups.
1113 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 14);