Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×C7×A4⋊C8 |
| unambiguous semidirect product | (C2×A4)⋊(C7×C8) |
| derived | A4 ⋊ C2×C7×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C7 | S4 |
| commutator, hypocenter | A4 | C2×C7×C8 |
| Frattini | C4 | C2×C7×S4 |
| Fitting | C23×C4×C7 | D3 |
| socle | C24×C7 | Dic3 |
| prime radical | C22×C7×A4 | C4 |
| cosocle | C4×A4 | C22×C7 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C4×C7 | |
178 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
280 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6417);