Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×D7×S4 |
| Fitting | C22×C4×C7 ⋊ C2×D3 |
| derived | C7×A4 ⋊ C22×C4 |
| Sylow | C7 ⋊ C2×C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D7×S4 |
| commutator, hypocenter | C7×A4 | C22×C4 |
| Frattini | C2 | C2×D7×S4 |
| Fitting | C22×C4×C7 | C2×D3 |
| socle | C23×C7 | C22×D3 |
| prime radical | C2×D7×S4 | C2 |
| cosocle | C2×C7×A4 | C23 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C4 | |
466 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | D7×AGL(1,𝔽5)×S4, C4×S4×AGL(1,𝔽7), D4×D7×S4 |
| maximal normal subgroup of | C4×S4×AGL(1,𝔽7), D4×D7×S4 |
| maximal quotient of | D7×AGL(1,𝔽5)×S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11326);