Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | D4×D7⋊A4 |
| unambiguous semidirect product | D7⋊(D4×A4) |
| unambiguous semidirect product | (D4×D7)⋊A4 |
| Fitting | C22×C7×D4 ⋊ C2×C3 |
| Hall | C22×D4×D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×D4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7⋊A4 |
| commutator, socle, cosocle | C23×C7 | C23×C3 |
| Fitting | C22×C7×D4 | C2×C3 |
| hypercenter | D4 | D7⋊A4 |
| hypocenter | C22×C7 | C2×C3×D4 |
| LCS | G ▷ C23×C7 ▷ C22×C7 | |
| UCS | C1 ◁ C2 ◁ D4 | |
634 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | S4×D7⋊A4, D4×A4×AGL(1,𝔽7) |
| maximal normal subgroup of | D4×A4×AGL(1,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11475);