Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#994 |
| Fitting, Hall | C2×C7×H(5,𝔽2) ⋊ C3 |
| Sylow | C2×H(5,𝔽2) ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C3×C7 |
| Frattini | C2 | C23×C7×A4 |
| Fitting | C2×C7×H(5,𝔽2) | C3 |
| hypercenter | C2×C7×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×Q8 | |
416 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11595);