Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×(C22⋊D7)⋊A4 |
| unambiguous semidirect product | (C2×C22⋊D7)⋊A4 |
| Fitting | C25×C7 ⋊ C2×C3 |
| Hall | C23×C22⋊D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×D4×A4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D7⋊A4 |
| commutator, cosocle | C23×C7 | C23×C3 |
| Frattini | C2 | C22×D7⋊A4 |
| Fitting | C25×C7 | C2×C3 |
| socle | C24×C7 | C22×C3 |
| hypercenter | C23 | D7⋊A4 |
| hypocenter | C22×C7 | C2×C3×D4 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
634 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11491);