Finite Group DatabasebetaFinite Group Databasebeta| order | 6678 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C53 · Schur multiplier (H2) = C1 |
| direct product | C2×C53×C7⋊C9 |
| unambiguous semidirect product | (C2×C7)⋊(C9×C53) |
| unambiguous semidirect product | (C2×C7×C53)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9×C53 |
| Hall | C2×C7×C53 ⋊ C9 |
| Hall | C7×C53 ⋊ C2×C9 |
| Hall | C2×C7 ⋊ C9×C53 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C53 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C9×C53 |
| Frattini | C3 | C2×ΓL(1,𝔽8)×C53 |
| Fitting, socle | C2×C3×C7×C53 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C53 | |
24 conjugacy classes of subgroups.
1590 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C53×C7⋊C9, C7⋊C9×Dic53, C53×Dic7⋊C9, … |
| maximal normal subgroup of | C4×C53×C7⋊C9, C7⋊C9×Dic53, C53×Dic7⋊C9, … |
| maximal quotient of | C4×C53×C7⋊C9, C22×C53×C7⋊C9, C2×C5×C53×C7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6678, 2);