Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | #67×AGL(1,𝔽8) |
| unambiguous semidirect product | (C23×#67)⋊C7 |
| Fitting | C25×C3 ⋊ C2×C7 |
| Hall | C23×#67 ⋊ C7 |
| Sylow | C3 ⋊ D4×AGL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×AGL(1,𝔽8) |
| commutator, socle, cosocle | C24×C3 | C22×C7 |
| Fitting | C25×C3 | C2×C7 |
| hypercenter | C22 | D3×AGL(1,𝔽8) |
| hypocenter | C23×C3 | C7×D4 |
| LCS | G ▷ C24×C3 ▷ C23×C3 | |
| UCS | C1 ◁ C2 ◁ C22 | |
254 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11315);