Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C7⋊C9×D53 |
| unambiguous semidirect product | C53⋊(C2×C7⋊C9) |
| unambiguous semidirect product | (C7×D53)⋊C9 |
| derived, Hall | C7×C53 ⋊ C2×C9 |
| Hall | C53×C7⋊C9 ⋊ C2 |
| Hall | C7×D53 ⋊ C9 |
| Sylow | C53 ⋊ C2×C7⋊C9 |
| Sylow | C7 ⋊ C9×D53 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | ΓL(1,𝔽8)×D53 |
| commutator, hypocenter | C7×C53 | C2×C9 |
| Fitting, socle | C3×C7×C53 | C2×C3 |
| LCS | G ▷ C7×C53 | |
| UCS | C1 ◁ C3 | |
24 conjugacy classes of subgroups.
420 conjugacy classes — view the full character table.
| maximal subgroup of | (C7⋊C9)⋊#6157, D53×D7⋊C9, C7⋊C9×#6157, … |
| maximal normal subgroup of | (C7⋊C9)⋊#6157, D53×D7⋊C9, C7⋊C9×#6157, … |
| maximal quotient of | C7⋊C9×Dic53, C2×C7⋊C9×D53, C5×C7⋊C9×D53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 10);