Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | #126×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×#126) |
| Fitting | C7×#126 ⋊ C2×C3 |
| Hall | D7×#126 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#126 |
| Sylow | C7 ⋊ C2×C3×#126 |
492 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#538, AGL(1,𝔽7)×#560, AGL(1,𝔽7)×#551, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#538, AGL(1,𝔽7)×#560, AGL(1,𝔽7)×#551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6727);