Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D7×#994 |
| Fitting | C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | D7×H(5,𝔽2) ⋊ C3 |
| Sylow | C7 ⋊ C2×#994 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×D7 |
| commutator, cosocle, hypocenter | C7×Q8 | C23×C3 |
| Fitting | C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C23×A4 |
| hypercenter | Q8 | A4×D7 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
416 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
95 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Ω+(4,𝔽3), AGL(1,𝔽7)×#994, #72246243, … |
| maximal normal subgroup of | D7×Ω+(4,𝔽3), AGL(1,𝔽7)×#994, #72246243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11465);