Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×D4×D21 |
| Fitting | C22×C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ C23×D4 |
| Sylow | C7 ⋊ C22×D3×D4 |
| Sylow | C3 ⋊ C22×D4×D7 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D21 |
| commutator, cosocle | C2×C3×C7 | C25 |
| Frattini | C2 | C24×D21 |
| Fitting | C22×C3×C7×D4 | C2 |
| socle | C23×C3×C7 | C23 |
| hypercenter | C22×D4 | D21 |
| hypocenter | C3×C7 | C23×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
2724 conjugacy classes of subgroups; only the 659 normal subgroups are shown.
This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 659 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11661);