Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#578⋊C3 |
| unambiguous semidirect product | #578⋊(C3×C7) |
| Fitting, Hall | C7×#578 ⋊ C3 |
| Sylow | #578 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D4×A4 |
| commutator, cosocle | #41 | C22×C3×C7 |
| Frattini | C4 | C22×C7×A4 |
| Fitting | C7×#578 | C3 |
| hypercenter | C7×D8 | A4 |
| hypocenter | Q8 | C3×C7×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×D8 | |
146 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10056);